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	|- U. On = On

Proof of Theorem unon

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eluni2 3584	. . . 4 |- ( x e. U. On <-> E. y e. On x e. y )
2		onelon 4121	. . . . 5 |- ( ( y e. On /\ x e. y ) -> x e. On )
3	2	rexlimiva 2428	. . . 4 |- ( E. y e. On x e. y -> x e. On )
4	1, 3	sylbi 114	. . 3 |- ( x e. U. On -> x e. On )
5		vex 2560	. . . . 5 |- x e. _V
6	5	sucid 4154	. . . 4 |- x e. suc x
7		suceloni 4227	. . . 4 |- ( x e. On -> suc x e. On )
8		elunii 3585	. . . 4 |- ( ( x e. suc x /\ suc x e. On ) -> x e. U. On )
9	6, 7, 8	sylancr 393	. . 3 |- ( x e. On -> x e. U. On )
10	4, 9	impbii 117	. 2 |- ( x e. U. On <-> x e. On )
11	10	eqriv 2037	1 |- U. On = On

Syntax hints:   = wceq 1243   e. wcel 1393   E.wrex 2307   U.cuni 3580   Oncon0 4100   suc 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