Theorem unon 4543

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 3839	. . . 4 |- ( x e. U. On <-> E. y e. On x e. y )
2		onelon 4415	. . . . 5 |- ( ( y e. On /\ x e. y ) -> x e. On )
3	2	rexlimiva 2606	. . . 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 2763	. . . . 5 |- x e. _V
6	5	sucid 4448	. . . 4 |- x e. suc x
7		onsuc 4533	. . . 4 |- ( x e. On -> suc x e. On )
8		elunii 3840	. . . 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 2190	1 |- U. On = On

Syntax hints:   = wceq 1364   e. wcel 2164   E.wrex 2473   U.cuni 3835   Oncon0 4394   suc csuc 4396

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-uni 3836  df-tr 4128  df-iord 4397  df-on 4399  df-suc 4402

This theorem is referenced by:  limon  4545  onintonm  4549  tfri1dALT  6404  rdgon  6439