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Theorem unon 4237
 Description: The class of all ordinal numbers is its own union. Exercise 11 of [TakeutiZaring] p. 40. (Contributed by NM, 12-Nov-2003.)
Assertion
Ref Expression
unon

Proof of Theorem unon
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eluni2 3584 . . . 4
2 onelon 4121 . . . . 5
32rexlimiva 2428 . . . 4
41, 3sylbi 114 . . 3
5 vex 2560 . . . . 5
65sucid 4154 . . . 4
7 suceloni 4227 . . . 4
8 elunii 3585 . . . 4
96, 7, 8sylancr 393 . . 3
104, 9impbii 117 . 2
1110eqriv 2037 1
 Colors of variables: wff set class Syntax hints:   wceq 1243   wcel 1393  wrex 2307  cuni 3580  con0 4100   csuc 4102 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-uni 3581  df-tr 3855  df-iord 4103  df-on 4105  df-suc 4108 This theorem is referenced by:  limon  4239  onintonm  4243
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