Theorem unon 4341

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 3663	. . . 4 |- ( x e. U. On <-> E. y e. On x e. y )
2		onelon 4220	. . . . 5 |- ( ( y e. On /\ x e. y ) -> x e. On )
3	2	rexlimiva 2485	. . . 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 2623	. . . . 5 |- x e. _V
6	5	sucid 4253	. . . 4 |- x e. suc x
7		suceloni 4331	. . . 4 |- ( x e. On -> suc x e. On )
8		elunii 3664	. . . 4 |- ( ( x e. suc x /\ suc x e. On ) -> x e. U. On )
9	6, 7, 8	sylancr 406	. . 3 |- ( x e. On -> x e. U. On )
10	4, 9	impbii 125	. 2 |- ( x e. U. On <-> x e. On )
11	10	eqriv 2086	1 |- U. On = On

Syntax hints:   = wceq 1290   e. wcel 1439   E.wrex 2361   U.cuni 3659   Oncon0 4199   suc csuc 4201

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269

This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-v 2622  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-uni 3660  df-tr 3943  df-iord 4202  df-on 4204  df-suc 4207

This theorem is referenced by:  limon  4343  onintonm  4347  tfri1dALT  6130  rdgon  6165