Theorem unon 4495

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 3800	. . . 4 |- ( x e. U. On <-> E. y e. On x e. y )
2		onelon 4369	. . . . 5 |- ( ( y e. On /\ x e. y ) -> x e. On )
3	2	rexlimiva 2582	. . . 4 |- ( E. y e. On x e. y -> x e. On )
4	1, 3	sylbi 120	. . 3 |- ( x e. U. On -> x e. On )
5		vex 2733	. . . . 5 |- x e. _V
6	5	sucid 4402	. . . 4 |- x e. suc x
7		suceloni 4485	. . . 4 |- ( x e. On -> suc x e. On )
8		elunii 3801	. . . 4 |- ( ( x e. suc x /\ suc x e. On ) -> x e. U. On )
9	6, 7, 8	sylancr 412	. . 3 |- ( x e. On -> x e. U. On )
10	4, 9	impbii 125	. 2 |- ( x e. U. On <-> x e. On )
11	10	eqriv 2167	1 |- U. On = On

Syntax hints:   = wceq 1348   e. wcel 2141   E.wrex 2449   U.cuni 3796   Oncon0 4348   suc csuc 4350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-uni 3797  df-tr 4088  df-iord 4351  df-on 4353  df-suc 4356

This theorem is referenced by:  limon  4497  onintonm  4501  tfri1dALT  6330  rdgon  6365