Theorem elvvv 5068

Description: Membership in universal class of ordered triples. (Contributed by NM, 17-Dec-2008.)

Assertion

Ref	Expression
elvvv	|- ( A e. ( ( _V X. _V ) X. _V ) <-> E. x E. y E. z A = <. <. x , y >. , z >. )

Distinct variable group:   x, y, z, A

Proof of Theorem elvvv

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elxp 5025	. 2 |- ( A e. ( ( _V X. _V ) X. _V ) <-> E. w E. z ( A = <. w , z >. /\ ( w e. ( _V X. _V ) /\ z e. _V ) ) )
2		anass 649	. . . . 5 |- ( ( ( A = <. w , z >. /\ w e. ( _V X. _V ) ) /\ z e. _V ) <-> ( A = <. w , z >. /\ ( w e. ( _V X. _V ) /\ z e. _V ) ) )
3		19.42vv 1778	. . . . . 6 |- ( E. x E. y ( A = <. w , z >. /\ w = <. x , y >. ) <-> ( A = <. w , z >. /\ E. x E. y w = <. x , y >. ) )
4		ancom 450	. . . . . . 7 |- ( ( w = <. x , y >. /\ A = <. w , z >. ) <-> ( A = <. w , z >. /\ w = <. x , y >. ) )
5	4	2exbii 1669	. . . . . 6 |- ( E. x E. y ( w = <. x , y >. /\ A = <. w , z >. ) <-> E. x E. y ( A = <. w , z >. /\ w = <. x , y >. ) )
6		vex 3112	. . . . . . . 8 |- z e. _V
7	6	biantru 505	. . . . . . 7 |- ( ( A = <. w , z >. /\ w e. ( _V X. _V ) ) <-> ( ( A = <. w , z >. /\ w e. ( _V X. _V ) ) /\ z e. _V ) )
8		elvv 5067	. . . . . . . 8 |- ( w e. ( _V X. _V ) <-> E. x E. y w = <. x , y >. )
9	8	anbi2i 694	. . . . . . 7 |- ( ( A = <. w , z >. /\ w e. ( _V X. _V ) ) <-> ( A = <. w , z >. /\ E. x E. y w = <. x , y >. ) )
10	7, 9	bitr3i 251	. . . . . 6 |- ( ( ( A = <. w , z >. /\ w e. ( _V X. _V ) ) /\ z e. _V ) <-> ( A = <. w , z >. /\ E. x E. y w = <. x , y >. ) )
11	3, 5, 10	3bitr4ri 278	. . . . 5 |- ( ( ( A = <. w , z >. /\ w e. ( _V X. _V ) ) /\ z e. _V ) <-> E. x E. y ( w = <. x , y >. /\ A = <. w , z >. ) )
12	2, 11	bitr3i 251	. . . 4 |- ( ( A = <. w , z >. /\ ( w e. ( _V X. _V ) /\ z e. _V ) ) <-> E. x E. y ( w = <. x , y >. /\ A = <. w , z >. ) )
13	12	2exbii 1669	. . 3 |- ( E. w E. z ( A = <. w , z >. /\ ( w e. ( _V X. _V ) /\ z e. _V ) ) <-> E. w E. z E. x E. y ( w = <. x , y >. /\ A = <. w , z >. ) )
14		exrot4 1854	. . . 4 |- ( E. x E. y E. w E. z ( w = <. x , y >. /\ A = <. w , z >. ) <-> E. w E. z E. x E. y ( w = <. x , y >. /\ A = <. w , z >. ) )
15		excom 1850	. . . . . 6 |- ( E. w E. z ( w = <. x , y >. /\ A = <. w , z >. ) <-> E. z E. w ( w = <. x , y >. /\ A = <. w , z >. ) )
16		opex 4720	. . . . . . . 8 |- <. x , y >. e. _V
17		opeq1 4219	. . . . . . . . 9 |- ( w = <. x , y >. -> <. w , z >. = <. <. x , y >. , z >. )
18	17	eqeq2d 2471	. . . . . . . 8 |- ( w = <. x , y >. -> ( A = <. w , z >. <-> A = <. <. x , y >. , z >. ) )
19	16, 18	ceqsexv 3146	. . . . . . 7 |- ( E. w ( w = <. x , y >. /\ A = <. w , z >. ) <-> A = <. <. x , y >. , z >. )
20	19	exbii 1668	. . . . . 6 |- ( E. z E. w ( w = <. x , y >. /\ A = <. w , z >. ) <-> E. z A = <. <. x , y >. , z >. )
21	15, 20	bitri 249	. . . . 5 |- ( E. w E. z ( w = <. x , y >. /\ A = <. w , z >. ) <-> E. z A = <. <. x , y >. , z >. )
22	21	2exbii 1669	. . . 4 |- ( E. x E. y E. w E. z ( w = <. x , y >. /\ A = <. w , z >. ) <-> E. x E. y E. z A = <. <. x , y >. , z >. )
23	14, 22	bitr3i 251	. . 3 |- ( E. w E. z E. x E. y ( w = <. x , y >. /\ A = <. w , z >. ) <-> E. x E. y E. z A = <. <. x , y >. , z >. )
24	13, 23	bitri 249	. 2 |- ( E. w E. z ( A = <. w , z >. /\ ( w e. ( _V X. _V ) /\ z e. _V ) ) <-> E. x E. y E. z A = <. <. x , y >. , z >. )
25	1, 24	bitri 249	1 |- ( A e. ( ( _V X. _V ) X. _V ) <-> E. x E. y E. z A = <. <. x , y >. , z >. )

Syntax hints:   <-> wb 184   /\ wa 369   = wceq 1395   E.wex 1613   e. wcel 1819   _Vcvv 3109   <.cop 4038   X. cxp 5006

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-opab 4516  df-xp 5014

This theorem is referenced by:  ssrelrel  5112  dftpos3  6991