Theorem unon 4435

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 3748	. . . 4 |- ( x e. U. On <-> E. y e. On x e. y )
2		onelon 4314	. . . . 5 |- ( ( y e. On /\ x e. y ) -> x e. On )
3	2	rexlimiva 2547	. . . 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 2692	. . . . 5 |- x e. _V
6	5	sucid 4347	. . . 4 |- x e. suc x
7		suceloni 4425	. . . 4 |- ( x e. On -> suc x e. On )
8		elunii 3749	. . . 4 |- ( ( x e. suc x /\ suc x e. On ) -> x e. U. On )
9	6, 7, 8	sylancr 411	. . 3 |- ( x e. On -> x e. U. On )
10	4, 9	impbii 125	. 2 |- ( x e. U. On <-> x e. On )
11	10	eqriv 2137	1 |- U. On = On

Syntax hints:   = wceq 1332   e. wcel 1481   E.wrex 2418   U.cuni 3744   Oncon0 4293   suc csuc 4295

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-uni 3745  df-tr 4035  df-iord 4296  df-on 4298  df-suc 4301

This theorem is referenced by:  limon  4437  onintonm  4441  tfri1dALT  6256  rdgon  6291