Theorem unon 4633

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 3918	. . . 4 |- ( x e. U. On <-> E. y e. On x e. y )
2		onelon 4505	. . . . 5 |- ( ( y e. On /\ x e. y ) -> x e. On )
3	2	rexlimiva 2655	. . . 4 |- ( E. y e. On x e. y -> x e. On )
4	1, 3	sylbi 121	. . 3 |- ( x e. U. On -> x e. On )
5		vex 2816	. . . . 5 |- x e. _V
6	5	sucid 4538	. . . 4 |- x e. suc x
7		onsuc 4623	. . . 4 |- ( x e. On -> suc x e. On )
8		elunii 3919	. . . 4 |- ( ( x e. suc x /\ suc x e. On ) -> x e. U. On )
9	6, 7, 8	sylancr 414	. . 3 |- ( x e. On -> x e. U. On )
10	4, 9	impbii 126	. 2 |- ( x e. U. On <-> x e. On )
11	10	eqriv 2229	1 |- U. On = On

Syntax hints:   = wceq 1398   e. wcel 2203   E.wrex 2521   U.cuni 3914   Oncon0 4484   suc csuc 4486

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-uni 3915  df-tr 4209  df-iord 4487  df-on 4489  df-suc 4492

This theorem is referenced by:  limon  4635  onintonm  4639  tfri1dALT  6582  rdgon  6617