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Theorem onzsl 3107
Description: An ordinal number is zero, a successor ordinal, or a limit ordinal number.
Assertion
Ref Expression
onzsl |- (A e. On <-> (A = (/) \/ E.x e. On A = suc x \/ (A e. V /\ Lim A)))
Distinct variable group:   x,A
Proof of Theorem onzsl
StepHypRef Expression
1 elisset 1808 . . 3 |- (A e. On -> A e. V)
21pm4.71ri 636 . 2 |- (A e. On <-> (A e. V /\ A e. On))
3 elong 2946 . . 3 |- (A e. V -> (A e. On <-> Ord A))
43pm5.32i 643 . 2 |- ((A e. V /\ A e. On) <-> (A e. V /\ Ord A))
5 andi 602 . . . 4 |- ((A e. V /\ ((A = (/) \/ E.x e. On A = suc x) \/ Lim A)) <-> ((A e. V /\ (A = (/) \/ E.x e. On A = suc x)) \/ (A e. V /\ Lim A)))
6 0ex 2701 . . . . . . . 8 |- (/) e. V
7 eleq1 1526 . . . . . . . 8 |- (A = (/) -> (A e. V <-> (/) e. V))
86, 7mpbiri 194 . . . . . . 7 |- (A = (/) -> A e. V)
9 visset 1804 . . . . . . . . . . 11 |- x e. V
109sucex 3040 . . . . . . . . . 10 |- suc x e. V
11 eleq1 1526 . . . . . . . . . 10 |- (A = suc x -> (A e. V <-> suc x e. V))
1210, 11mpbiri 194 . . . . . . . . 9 |- (A = suc x -> A e. V)
1312a1i 8 . . . . . . . 8 |- (x e. On -> (A = suc x -> A e. V))
1413r19.23aiv 1735 . . . . . . 7 |- (E.x e. On A = suc x -> A e. V)
158, 14jaoi 341 . . . . . 6 |- ((A = (/) \/ E.x e. On A = suc x) -> A e. V)
1615pm4.71ri 636 . . . . 5 |- ((A = (/) \/ E.x e. On A = suc x) <-> (A e. V /\ (A = (/) \/ E.x e. On A = suc x)))
1716orbi1i 256 . . . 4 |- (((A = (/) \/ E.x e. On A = suc x) \/ (A e. V /\ Lim A)) <-> ((A e. V /\ (A = (/) \/ E.x e. On A = suc x)) \/ (A e. V /\ Lim A)))
185, 17bitr4 176 . . 3 |- ((A e. V /\ ((A = (/) \/ E.x e. On A = suc x) \/ Lim A)) <-> ((A = (/) \/ E.x e. On A = suc x) \/ (A e. V /\ Lim A)))
19 ordzsl 3106 . . . . 5 |- (Ord A <-> (A = (/) \/ E.x e. On A = suc x \/ Lim A))
20 df-3or 774 . . . . 5 |- ((A = (/) \/ E.x e. On A = suc x \/ Lim A) <-> ((A = (/) \/ E.x e. On A = suc x) \/ Lim A))
2119, 20bitr 173 . . . 4 |- (Ord A <-> ((A = (/) \/ E.x e. On A = suc x) \/ Lim A))
2221anbi2i 479 . . 3 |- ((A e. V /\ Ord A) <-> (A e. V /\ ((A = (/) \/ E.x e. On A = suc x) \/ Lim A)))
23 df-3or 774 . . 3 |- ((A = (/) \/ E.x e. On A = suc x \/ (A e. V /\ Lim A)) <-> ((A = (/) \/ E.x e. On A = suc x) \/ (A e. V /\ Lim A)))
2418, 22, 233bitr4 183 . 2 |- ((A e. V /\ Ord A) <-> (A = (/) \/ E.x e. On A = suc x \/ (A e. V /\ Lim A)))
252, 4, 243bitr 177 1 |- (A e. On <-> (A = (/) \/ E.x e. On A = suc x \/ (A e. V /\ Lim A)))
Colors of variables: wff set class
Syntax hints:   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   \/ w3o 772   = wceq 953   e. wcel 955  E.wrex 1638  Vcvv 1802  (/)c0 2270  Ord word 2937  Oncon0 2938  Lim wlim 2939  suc csuc 2940
This theorem is referenced by:  oawordeulem 4172  r1val1 4630
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-nul 2700  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-if 2352  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-tr 2671  df-eprel 2821  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944
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