Theorem unon 4488

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 3793	. . . 4 |- ( x e. U. On <-> E. y e. On x e. y )
2		onelon 4362	. . . . 5 |- ( ( y e. On /\ x e. y ) -> x e. On )
3	2	rexlimiva 2578	. . . 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 2729	. . . . 5 |- x e. _V
6	5	sucid 4395	. . . 4 |- x e. suc x
7		suceloni 4478	. . . 4 |- ( x e. On -> suc x e. On )
8		elunii 3794	. . . 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 2162	1 |- U. On = On

Syntax hints:   = wceq 1343   e. wcel 2136   E.wrex 2445   U.cuni 3789   Oncon0 4341   suc csuc 4343

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-uni 3790  df-tr 4081  df-iord 4344  df-on 4346  df-suc 4349

This theorem is referenced by:  limon  4490  onintonm  4494  tfri1dALT  6319  rdgon  6354