Theorem unon 4427

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 3740	. . . 4 |- ( x e. U. On <-> E. y e. On x e. y )
2		onelon 4306	. . . . 5 |- ( ( y e. On /\ x e. y ) -> x e. On )
3	2	rexlimiva 2544	. . . 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 2689	. . . . 5 |- x e. _V
6	5	sucid 4339	. . . 4 |- x e. suc x
7		suceloni 4417	. . . 4 |- ( x e. On -> suc x e. On )
8		elunii 3741	. . . 4 |- ( ( x e. suc x /\ suc x e. On ) -> x e. U. On )
9	6, 7, 8	sylancr 410	. . 3 |- ( x e. On -> x e. U. On )
10	4, 9	impbii 125	. 2 |- ( x e. U. On <-> x e. On )
11	10	eqriv 2136	1 |- U. On = On

Syntax hints:   = wceq 1331   e. wcel 1480   E.wrex 2417   U.cuni 3736   Oncon0 4285   suc csuc 4287

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-uni 3737  df-tr 4027  df-iord 4288  df-on 4290  df-suc 4293

This theorem is referenced by:  limon  4429  onintonm  4433  tfri1dALT  6248  rdgon  6283