Theorem unon 4653

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 3934	. . . 4 |- ( x e. U. On <-> E. y e. On x e. y )
2		onelon 4524	. . . . 5 |- ( ( y e. On /\ x e. y ) -> x e. On )
3	2	rexlimiva 2663	. . . 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 2824	. . . . 5 |- x e. _V
6	5	sucid 4557	. . . 4 |- x e. suc x
7		onsuc 4643	. . . 4 |- ( x e. On -> suc x e. On )
8		elunii 3935	. . . 4 |- ( ( x e. suc x /\ suc x e. On ) -> x e. U. On )
9	6, 7, 8	sylancr 418	. . 3 |- ( x e. On -> x e. U. On )
10	4, 9	impbii 126	. 2 |- ( x e. U. On <-> x e. On )
11	10	eqriv 2235	1 |- U. On = On

Syntax hints:   = wceq 1402   e. wcel 2209   E.wrex 2529   U.cuni 3930   Oncon0 4503   suc csuc 4505

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-uni 3931  df-tr 4225  df-iord 4506  df-on 4508  df-suc 4511

This theorem is referenced by:  limon  4655  onintonm  4659  tfri1dALT  6612  rdgon  6647