Theorem elvvv 4913

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 4870	. 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 659	. . . . 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 1847	. . . . . 6 |- ( E. x E. y ( A = <. w , z >. /\ w = <. x , y >. ) <-> ( A = <. w , z >. /\ E. x E. y w = <. x , y >. ) )
4		ancom 456	. . . . . . 7 |- ( ( w = <. x , y >. /\ A = <. w , z >. ) <-> ( A = <. w , z >. /\ w = <. x , y >. ) )
5	4	2exbii 1730	. . . . . 6 |- ( E. x E. y ( w = <. x , y >. /\ A = <. w , z >. ) <-> E. x E. y ( A = <. w , z >. /\ w = <. x , y >. ) )
6		vex 3060	. . . . . . . 8 |- z e. _V
7	6	biantru 512	. . . . . . 7 |- ( ( A = <. w , z >. /\ w e. ( _V X. _V ) ) <-> ( ( A = <. w , z >. /\ w e. ( _V X. _V ) ) /\ z e. _V ) )
8		elvv 4912	. . . . . . . 8 |- ( w e. ( _V X. _V ) <-> E. x E. y w = <. x , y >. )
9	8	anbi2i 705	. . . . . . 7 |- ( ( A = <. w , z >. /\ w e. ( _V X. _V ) ) <-> ( A = <. w , z >. /\ E. x E. y w = <. x , y >. ) )
10	7, 9	bitr3i 259	. . . . . 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 286	. . . . 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 259	. . . 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 1730	. . 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 1943	. . . 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 1938	. . . . . 6 |- ( E. w E. z ( w = <. x , y >. /\ A = <. w , z >. ) <-> E. z E. w ( w = <. x , y >. /\ A = <. w , z >. ) )
16		opex 4678	. . . . . . . 8 |- <. x , y >. e. _V
17		opeq1 4180	. . . . . . . . 9 |- ( w = <. x , y >. -> <. w , z >. = <. <. x , y >. , z >. )
18	17	eqeq2d 2472	. . . . . . . 8 |- ( w = <. x , y >. -> ( A = <. w , z >. <-> A = <. <. x , y >. , z >. ) )
19	16, 18	ceqsexv 3096	. . . . . . 7 |- ( E. w ( w = <. x , y >. /\ A = <. w , z >. ) <-> A = <. <. x , y >. , z >. )
20	19	exbii 1729	. . . . . 6 |- ( E. z E. w ( w = <. x , y >. /\ A = <. w , z >. ) <-> E. z A = <. <. x , y >. , z >. )
21	15, 20	bitri 257	. . . . 5 |- ( E. w E. z ( w = <. x , y >. /\ A = <. w , z >. ) <-> E. z A = <. <. x , y >. , z >. )
22	21	2exbii 1730	. . . 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 259	. . 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 257	. 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 257	1 |- ( A e. ( ( _V X. _V ) X. _V ) <-> E. x E. y E. z A = <. <. x , y >. , z >. )

Syntax hints:   <-> wb 189   /\ wa 375   = wceq 1455   E.wex 1674   e. wcel 1898   _Vcvv 3057   <.cop 3986   X. cxp 4851

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4539  ax-nul 4548  ax-pr 4653

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-rab 2758  df-v 3059  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-opab 4476  df-xp 4859

This theorem is referenced by:  ssrelrel  4954  dftpos3  7017