Theorem unon 4255

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 3605	. . . 4 |- ( x e. U. On <-> E. y e. On x e. y )
2		onelon 4139	. . . . 5 |- ( ( y e. On /\ x e. y ) -> x e. On )
3	2	rexlimiva 2472	. . . 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 2604	. . . . 5 |- x e. _V
6	5	sucid 4172	. . . 4 |- x e. suc x
7		suceloni 4245	. . . 4 |- ( x e. On -> suc x e. On )
8		elunii 3606	. . . 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 2078	1 |- U. On = On

Syntax hints:   = wceq 1284   e. wcel 1433   E.wrex 2349   U.cuni 3601   Oncon0 4118   suc csuc 4120

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-uni 3602  df-tr 3876  df-iord 4121  df-on 4123  df-suc 4126

This theorem is referenced by:  limon  4257  onintonm  4261