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Theorem unon 6666
 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 4253 . . . 4
2 onelon 4908 . . . . 5
32rexlimiva 2945 . . . 4
41, 3sylbi 195 . . 3
5 vex 3112 . . . . 5
65sucid 4962 . . . 4
7 suceloni 6648 . . . 4
8 elunii 4254 . . . 4
96, 7, 8sylancr 663 . . 3
104, 9impbii 188 . 2
1110eqriv 2453 1
 Colors of variables: wff setvar class Syntax hints:  =wceq 1395  e.wcel 1818  E.wrex 2808  U.cuni 4249   con0 4883  succsuc 4885 This theorem is referenced by:  ordunisuc  6667  limon  6671  orduninsuc  6678  ordtoplem  29900  ordcmp  29912 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691  ax-un 6592 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-pss 3491  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-tp 4034  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-tr 4546  df-eprel 4796  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-suc 4889
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