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Theorem cda1en 4898
Description: Cardinal addition with cardinal one (which is the same as ordinal one). Used in proof of Theorem 6J of [Enderton] p. 143.
Hypothesis
Ref Expression
cda0en.1 |- A e. V
Assertion
Ref Expression
cda1en |- (A +c 1o) ~~ suc (card` A)
Proof of Theorem cda1en
StepHypRef Expression
1 cda0en.1 . . . . 5 |- A e. V
2 0ex 2701 . . . . . 6 |- (/) e. V
31, 2xpsnen 4415 . . . . 5 |- (A X. {(/)}) ~~ A
4 cardid 4800 . . . . 5 |- (card` A) ~~ A
51, 3, 4entr4 4400 . . . 4 |- (A X. {(/)}) ~~ (card` A)
6 1on 4122 . . . . . 6 |- 1o e. On
76elisseti 1809 . . . . 5 |- 1o e. V
87, 7xpsnen 4415 . . . . 5 |- (1o X. {1o}) ~~ 1o
9 fvex 3717 . . . . . 6 |- (card` A) e. V
109ensn1 4405 . . . . 5 |- {(card` A)} ~~ 1o
117, 8, 10entr4 4400 . . . 4 |- (1o X. {1o}) ~~ {(card` A)}
125, 11pm3.2i 285 . . 3 |- ((A X. {(/)}) ~~ (card` A) /\ (1o X. {1o}) ~~ {(card` A)})
13 xp01disj 4127 . . . 4 |- ((A X. {(/)}) i^i (1o X. {1o})) = (/)
14 cardon 4799 . . . . . 6 |- (card` A) e. On
1514onord 3085 . . . . 5 |- Ord (card` A)
16 orddisj 2975 . . . . 5 |- (Ord (card` A) -> ((card` A) i^i {(card` A)}) = (/))
1715, 16ax-mp 7 . . . 4 |- ((card` A) i^i {(card` A)}) = (/)
1813, 17pm3.2i 285 . . 3 |- (((A X. {(/)}) i^i (1o X. {1o})) = (/) /\ ((card` A) i^i {(card` A)}) = (/))
19 unen 4414 . . 3 |- ((((A X. {(/)}) ~~ (card` A) /\ (1o X. {1o}) ~~ {(card` A)}) /\ (((A X. {(/)}) i^i (1o X. {1o})) = (/) /\ ((card` A) i^i {(card` A)}) = (/))) -> ((A X. {(/)}) u. (1o X. {1o})) ~~ ((card` A) u. {(card` A)}))
2012, 18, 19mp2an 695 . 2 |- ((A X. {(/)}) u. (1o X. {1o})) ~~ ((card` A) u. {(card` A)})
211, 7cdaval 4892 . 2 |- (A +c 1o) = ((A X. {(/)}) u. (1o X. {1o}))
22 df-suc 2944 . 2 |- suc (card` A) = ((card` A) u. {(card` A)})
2320, 21, 223brtr4 2633 1 |- (A +c 1o) ~~ suc (card` A)
Colors of variables: wff set class
Syntax hints:   /\ wa 223   = wceq 953   e. wcel 955  Vcvv 1802   u. cun 2035   i^i cin 2036  (/)c0 2270  {csn 2399   class class class wbr 2609  Ord word 2937  Oncon0 2938  suc csuc 2940   X. cxp 3158  ` cfv 3172  (class class class)co 3948  1oc1o 4112   ~~ cen 4348  cardccrd 4785   +c ccda 4889
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-9 962  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-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-ac 4716
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-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  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-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-suc 2944  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-opr 3950  df-oprab 3951  df-1o 4117  df-er 4245  df-en 4351  df-card 4788  df-cda 4890
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