Theorem unon 4326

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 3655	. . . 4 |- ( x e. U. On <-> E. y e. On x e. y )
2		onelon 4209	. . . . 5 |- ( ( y e. On /\ x e. y ) -> x e. On )
3	2	rexlimiva 2484	. . . 4 |- ( E. y e. On x e. y -> x e. On )
4	1, 3	sylbi 119	. . 3 |- ( x e. U. On -> x e. On )
5		vex 2622	. . . . 5 |- x e. _V
6	5	sucid 4242	. . . 4 |- x e. suc x
7		suceloni 4316	. . . 4 |- ( x e. On -> suc x e. On )
8		elunii 3656	. . . 4 |- ( ( x e. suc x /\ suc x e. On ) -> x e. U. On )
9	6, 7, 8	sylancr 405	. . 3 |- ( x e. On -> x e. U. On )
10	4, 9	impbii 124	. 2 |- ( x e. U. On <-> x e. On )
11	10	eqriv 2085	1 |- U. On = On

Syntax hints:   = wceq 1289   e. wcel 1438   E.wrex 2360   U.cuni 3651   Oncon0 4188   suc csuc 4190

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3955  ax-pow 4007  ax-pr 4034  ax-un 4258

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3429  df-sn 3450  df-pr 3451  df-uni 3652  df-tr 3935  df-iord 4191  df-on 4193  df-suc 4196

This theorem is referenced by:  limon  4328  onintonm  4332  tfri1dALT  6108  rdgon  6143