Theorem unon 4577

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 3868	. . . 4 |- ( x e. U. On <-> E. y e. On x e. y )
2		onelon 4449	. . . . 5 |- ( ( y e. On /\ x e. y ) -> x e. On )
3	2	rexlimiva 2620	. . . 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 2779	. . . . 5 |- x e. _V
6	5	sucid 4482	. . . 4 |- x e. suc x
7		onsuc 4567	. . . 4 |- ( x e. On -> suc x e. On )
8		elunii 3869	. . . 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 2204	1 |- U. On = On

Syntax hints:   = wceq 1373   e. wcel 2178   E.wrex 2487   U.cuni 3864   Oncon0 4428   suc csuc 4430

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-uni 3865  df-tr 4159  df-iord 4431  df-on 4433  df-suc 4436

This theorem is referenced by:  limon  4579  onintonm  4583  tfri1dALT  6460  rdgon  6495