Theorem unon 7031

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 4440	. . . 4 |- ( x e. U. On <-> E. y e. On x e. y )
2		onelon 5748	. . . . 5 |- ( ( y e. On /\ x e. y ) -> x e. On )
3	2	rexlimiva 3028	. . . 4 |- ( E. y e. On x e. y -> x e. On )
4	1, 3	sylbi 207	. . 3 |- ( x e. U. On -> x e. On )
5		vex 3203	. . . . 5 |- x e. _V
6	5	sucid 5804	. . . 4 |- x e. suc x
7		suceloni 7013	. . . 4 |- ( x e. On -> suc x e. On )
8		elunii 4441	. . . 4 |- ( ( x e. suc x /\ suc x e. On ) -> x e. U. On )
9	6, 7, 8	sylancr 695	. . 3 |- ( x e. On -> x e. U. On )
10	4, 9	impbii 199	. 2 |- ( x e. U. On <-> x e. On )
11	10	eqriv 2619	1 |- U. On = On

Syntax hints:   = wceq 1483   e. wcel 1990   E.wrex 2913   U.cuni 4436   Oncon0 5723   suc csuc 5725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727  df-suc 5729

This theorem is referenced by:  ordunisuc  7032  limon  7036  orduninsuc  7043  ordtoplem  32434  ordcmp  32446