Theorem unon 4559

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 3854	. . . 4 |- ( x e. U. On <-> E. y e. On x e. y )
2		onelon 4431	. . . . 5 |- ( ( y e. On /\ x e. y ) -> x e. On )
3	2	rexlimiva 2618	. . . 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 2775	. . . . 5 |- x e. _V
6	5	sucid 4464	. . . 4 |- x e. suc x
7		onsuc 4549	. . . 4 |- ( x e. On -> suc x e. On )
8		elunii 3855	. . . 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 2202	1 |- U. On = On

Syntax hints:   = wceq 1373   e. wcel 2176   E.wrex 2485   U.cuni 3850   Oncon0 4410   suc csuc 4412

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-uni 3851  df-tr 4143  df-iord 4413  df-on 4415  df-suc 4418

This theorem is referenced by:  limon  4561  onintonm  4565  tfri1dALT  6437  rdgon  6472