Theorem elvvuni 5179

Description: An ordered pair contains its union. (Contributed by NM, 16-Sep-2006.)

Assertion

Ref	Expression
elvvuni	|- ( A e. ( _V X. _V ) -> U. A e. A )

Proof of Theorem elvvuni

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elvv 5177	. 2 |- ( A e. ( _V X. _V ) <-> E. x E. y A = <. x , y >. )
2		vex 3203	. . . . . 6 |- x e. _V
3		vex 3203	. . . . . 6 |- y e. _V
4	2, 3	uniop 4977	. . . . 5 |- U. <. x , y >. = { x , y }
5	2, 3	opi2 4938	. . . . 5 |- { x , y } e. <. x , y >.
6	4, 5	eqeltri 2697	. . . 4 |- U. <. x , y >. e. <. x , y >.
7		unieq 4444	. . . . 5 |- ( A = <. x , y >. -> U. A = U. <. x , y >. )
8		id 22	. . . . 5 |- ( A = <. x , y >. -> A = <. x , y >. )
9	7, 8	eleq12d 2695	. . . 4 |- ( A = <. x , y >. -> ( U. A e. A <-> U. <. x , y >. e. <. x , y >. ) )
10	6, 9	mpbiri 248	. . 3 |- ( A = <. x , y >. -> U. A e. A )
11	10	exlimivv 1860	. 2 |- ( E. x E. y A = <. x , y >. -> U. A e. A )
12	1, 11	sylbi 207	1 |- ( A e. ( _V X. _V ) -> U. A e. A )

Syntax hints:   -> wi 4   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   {cpr 4179   <.cop 4183   U.cuni 4436   X. cxp 5112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-opab 4713  df-xp 5120

This theorem is referenced by:  unielxp  7204