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Theorem weinxp 3223
Description: Intersection of well-ordering with cross product of its field.
Assertion
Ref Expression
weinxp |- (R We A <-> (R i^i (A X. A)) We A)
Proof of Theorem weinxp
StepHypRef Expression
1 ssel 2053 . . . . . . . . . . . . . 14 |- (z (_ A -> (x e. z -> x e. A))
2 ssel 2053 . . . . . . . . . . . . . 14 |- (z (_ A -> (y e. z -> y e. A))
31, 2anim12d 556 . . . . . . . . . . . . 13 |- (z (_ A -> ((x e. z /\ y e. z) -> (x e. A /\ y e. A)))
4 brinxp 3222 . . . . . . . . . . . . . 14 |- ((y e. A /\ x e. A) -> (yRx <-> y(R i^i (A X. A))x))
54ancoms 436 . . . . . . . . . . . . 13 |- ((x e. A /\ y e. A) -> (yRx <-> y(R i^i (A X. A))x))
63, 5syl6 22 . . . . . . . . . . . 12 |- (z (_ A -> ((x e. z /\ y e. z) -> (yRx <-> y(R i^i (A X. A))x)))
76exp3a 375 . . . . . . . . . . 11 |- (z (_ A -> (x e. z -> (y e. z -> (yRx <-> y(R i^i (A X. A))x))))
87imp31 362 . . . . . . . . . 10 |- (((z (_ A /\ x e. z) /\ y e. z) -> (yRx <-> y(R i^i (A X. A))x))
98negbid 609 . . . . . . . . 9 |- (((z (_ A /\ x e. z) /\ y e. z) -> (-. yRx <-> -. y(R i^i (A X. A))x))
109ralbidva 1651 . . . . . . . 8 |- ((z (_ A /\ x e. z) -> (A.y e. z -. yRx <-> A.y e. z -. y(R i^i (A X. A))x))
1110rexbidva 1652 . . . . . . 7 |- (z (_ A -> (E.x e. z A.y e. z -. yRx <-> E.x e. z A.y e. z -. y(R i^i (A X. A))x))
1211adantr 389 . . . . . 6 |- ((z (_ A /\ z =/= (/)) -> (E.x e. z A.y e. z -. yRx <-> E.x e. z A.y e. z -. y(R i^i (A X. A))x))
1312pm5.74i 582 . . . . 5 |- (((z (_ A /\ z =/= (/)) -> E.x e. z A.y e. z -. yRx) <-> ((z (_ A /\ z =/= (/)) -> E.x e. z A.y e. z -. y(R i^i (A X. A))x))
1413albii 996 . . . 4 |- (A.z((z (_ A /\ z =/= (/)) -> E.x e. z A.y e. z -. yRx) <-> A.z((z (_ A /\ z =/= (/)) -> E.x e. z A.y e. z -. y(R i^i (A X. A))x))
15 df-fr 2907 . . . 4 |- (R Fr A <-> A.z((z (_ A /\ z =/= (/)) -> E.x e. z A.y e. z -. yRx))
16 df-fr 2907 . . . 4 |- ((R i^i (A X. A)) Fr A <-> A.z((z (_ A /\ z =/= (/)) -> E.x e. z A.y e. z -. y(R i^i (A X. A))x))
1714, 15, 163bitr4 183 . . 3 |- (R Fr A <-> (R i^i (A X. A)) Fr A)
18 brinxp 3222 . . . . . . 7 |- ((x e. A /\ y e. A) -> (xRy <-> x(R i^i (A X. A))y))
19 pm4.2i 171 . . . . . . 7 |- ((x e. A /\ y e. A) -> (x = y <-> x = y))
2018, 19, 53orbi123d 889 . . . . . 6 |- ((x e. A /\ y e. A) -> ((xRy \/ x = y \/ yRx) <-> (x(R i^i (A X. A))y \/ x = y \/ y(R i^i (A X. A))x)))
2120pm5.74i 582 . . . . 5 |- (((x e. A /\ y e. A) -> (xRy \/ x = y \/ yRx)) <-> ((x e. A /\ y e. A) -> (x(R i^i (A X. A))y \/ x = y \/ y(R i^i (A X. A))x)))
22212albii 997 . . . 4 |- (A.xA.y((x e. A /\ y e. A) -> (xRy \/ x = y \/ yRx)) <-> A.xA.y((x e. A /\ y e. A) -> (x(R i^i (A X. A))y \/ x = y \/ y(R i^i (A X. A))x)))
23 r2al 1668 . . . 4 |- (A.x e. A A.y e. A (xRy \/ x = y \/ yRx) <-> A.xA.y((x e. A /\ y e. A) -> (xRy \/ x = y \/ yRx)))
24 r2al 1668 . . . 4 |- (A.x e. A A.y e. A (x(R i^i (A X. A))y \/ x = y \/ y(R i^i (A X. A))x) <-> A.xA.y((x e. A /\ y e. A) -> (x(R i^i (A X. A))y \/ x = y \/ y(R i^i (A X. A))x)))
2522, 23, 243bitr4 183 . . 3 |- (A.x e. A A.y e. A (xRy \/ x = y \/ yRx) <-> A.x e. A A.y e. A (x(R i^i (A X. A))y \/ x = y \/ y(R i^i (A X. A))x))
2617, 25anbi12i 481 . 2 |- ((R Fr A /\ A.x e. A A.y e. A (xRy \/ x = y \/ yRx)) <-> ((R i^i (A X. A)) Fr A /\ A.x e. A A.y e. A (x(R i^i (A X. A))y \/ x = y \/ y(R i^i (A X. A))x)))
27 dfwe2 2925 . 2 |- (R We A <-> (R Fr A /\ A.x e. A A.y e. A (xRy \/ x = y \/ yRx)))
28 dfwe2 2925 . 2 |- ((R i^i (A X. A)) We A <-> ((R i^i (A X. A)) Fr A /\ A.x e. A A.y e. A (x(R i^i (A X. A))y \/ x = y \/ y(R i^i (A X. A))x)))
2926, 27, 283bitr4 183 1 |- (R We A <-> (R i^i (A X. A)) We A)
Colors of variables: wff set class
Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223   \/ w3o 772  A.wal 951   = wceq 953   e. wcel 955   =/= wne 1577  A.wral 1637  E.wrex 1638   i^i cin 2036   (_ wss 2037  (/)c0 2270   class class class wbr 2609   Fr wfr 2905   We wwe 2906   X. cxp 3158
This theorem is referenced by:  weth 4759
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857
This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-xp 3174
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