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Theorem unon 4202
 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 On = On

Proof of Theorem unon
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eluni2 3575 . . . 4 (x On ↔ y On x y)
2 onelon 4087 . . . . 5 ((y On x y) → x On)
32rexlimiva 2422 . . . 4 (y On x yx On)
41, 3sylbi 114 . . 3 (x On → x On)
5 vex 2554 . . . . 5 x V
65sucid 4120 . . . 4 x suc x
7 suceloni 4193 . . . 4 (x On → suc x On)
8 elunii 3576 . . . 4 ((x suc x suc x On) → x On)
96, 7, 8sylancr 393 . . 3 (x On → x On)
104, 9impbii 117 . 2 (x On ↔ x On)
1110eqriv 2034 1 On = On
 Colors of variables: wff set class Syntax hints:   = wceq 1242   ∈ wcel 1390  ∃wrex 2301  ∪ cuni 3571  Oncon0 4066  suc csuc 4068 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-uni 3572  df-tr 3846  df-iord 4069  df-on 4071  df-suc 4074 This theorem is referenced by:  limon  4204
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