Theorem elvvv 5059

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 5016	. 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 1951	. . . . . 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 1645	. . . . . 6 |- ( E. x E. y ( w = <. x , y >. /\ A = <. w , z >. ) <-> E. x E. y ( A = <. w , z >. /\ w = <. x , y >. ) )
6		vex 3116	. . . . . . . 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 5058	. . . . . . . 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 1645	. . 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 1802	. . . 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 1798	. . . . . 6 |- ( E. w E. z ( w = <. x , y >. /\ A = <. w , z >. ) <-> E. z E. w ( w = <. x , y >. /\ A = <. w , z >. ) )
16		opex 4711	. . . . . . . 8 |- <. x , y >. e. _V
17		opeq1 4213	. . . . . . . . 9 |- ( w = <. x , y >. -> <. w , z >. = <. <. x , y >. , z >. )
18	17	eqeq2d 2481	. . . . . . . 8 |- ( w = <. x , y >. -> ( A = <. w , z >. <-> A = <. <. x , y >. , z >. ) )
19	16, 18	ceqsexv 3150	. . . . . . 7 |- ( E. w ( w = <. x , y >. /\ A = <. w , z >. ) <-> A = <. <. x , y >. , z >. )
20	19	exbii 1644	. . . . . 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 1645	. . . 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 1379   E.wex 1596   e. wcel 1767   _Vcvv 3113   <.cop 4033   X. cxp 4997

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-opab 4506  df-xp 5005

This theorem is referenced by:  ssrelrel  5103  dftpos3  6974