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Theorem unon 4182
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 3554 . . . 4 (x On ↔ y On x y)
2 onelon 4066 . . . . 5 ((y On x y) → x On)
32rexlimiva 2402 . . . 4 (y On x yx On)
41, 3sylbi 114 . . 3 (x On → x On)
5 vex 2534 . . . . 5 x V
65sucid 4099 . . . 4 x suc x
7 suceloni 4173 . . . 4 (x On → suc x On)
8 elunii 3555 . . . 4 ((x suc x suc x On) → x On)
96, 7, 8sylancr 395 . . 3 (x On → x On)
104, 9impbii 117 . 2 (x On ↔ x On)
1110eqriv 2015 1 On = On
Colors of variables: wff set class
Syntax hints:   = wceq 1226   wcel 1370  wrex 2281   cuni 3550  Oncon0 4045  suc csuc 4047
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 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-13 1381  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914  ax-un 4116
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-uni 3551  df-tr 3825  df-iord 4048  df-on 4050  df-suc 4053
This theorem is referenced by:  limon  4184
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