Theorem elvvv 5178

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 5131	. 2 |- ( A e. ( ( _V X. _V ) X. _V ) <-> E. w E. z ( A = <. w , z >. /\ ( w e. ( _V X. _V ) /\ z e. _V ) ) )
2		ancom 466	. . . . . 6 |- ( ( w = <. x , y >. /\ A = <. w , z >. ) <-> ( A = <. w , z >. /\ w = <. x , y >. ) )
3	2	2exbii 1775	. . . . 5 |- ( E. x E. y ( w = <. x , y >. /\ A = <. w , z >. ) <-> E. x E. y ( A = <. w , z >. /\ w = <. x , y >. ) )
4		19.42vv 1920	. . . . . 6 |- ( E. x E. y ( A = <. w , z >. /\ w = <. x , y >. ) <-> ( A = <. w , z >. /\ E. x E. y w = <. x , y >. ) )
5		elvv 5177	. . . . . . 7 |- ( w e. ( _V X. _V ) <-> E. x E. y w = <. x , y >. )
6	5	anbi2i 730	. . . . . 6 |- ( ( A = <. w , z >. /\ w e. ( _V X. _V ) ) <-> ( A = <. w , z >. /\ E. x E. y w = <. x , y >. ) )
7		vex 3203	. . . . . . 7 |- z e. _V
8	7	biantru 526	. . . . . 6 |- ( ( A = <. w , z >. /\ w e. ( _V X. _V ) ) <-> ( ( A = <. w , z >. /\ w e. ( _V X. _V ) ) /\ z e. _V ) )
9	4, 6, 8	3bitr2i 288	. . . . 5 |- ( E. x E. y ( A = <. w , z >. /\ w = <. x , y >. ) <-> ( ( A = <. w , z >. /\ w e. ( _V X. _V ) ) /\ z e. _V ) )
10		anass 681	. . . . 5 |- ( ( ( A = <. w , z >. /\ w e. ( _V X. _V ) ) /\ z e. _V ) <-> ( A = <. w , z >. /\ ( w e. ( _V X. _V ) /\ z e. _V ) ) )
11	3, 9, 10	3bitrri 287	. . . 4 |- ( ( A = <. w , z >. /\ ( w e. ( _V X. _V ) /\ z e. _V ) ) <-> E. x E. y ( w = <. x , y >. /\ A = <. w , z >. ) )
12	11	2exbii 1775	. . 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 >. ) )
13		exrot4 2046	. . 3 |- ( 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 >. ) )
14		excom 2042	. . . . 5 |- ( E. w E. z ( w = <. x , y >. /\ A = <. w , z >. ) <-> E. z E. w ( w = <. x , y >. /\ A = <. w , z >. ) )
15		opex 4932	. . . . . . 7 |- <. x , y >. e. _V
16		opeq1 4402	. . . . . . . 8 |- ( w = <. x , y >. -> <. w , z >. = <. <. x , y >. , z >. )
17	16	eqeq2d 2632	. . . . . . 7 |- ( w = <. x , y >. -> ( A = <. w , z >. <-> A = <. <. x , y >. , z >. ) )
18	15, 17	ceqsexv 3242	. . . . . 6 |- ( E. w ( w = <. x , y >. /\ A = <. w , z >. ) <-> A = <. <. x , y >. , z >. )
19	18	exbii 1774	. . . . 5 |- ( E. z E. w ( w = <. x , y >. /\ A = <. w , z >. ) <-> E. z A = <. <. x , y >. , z >. )
20	14, 19	bitri 264	. . . 4 |- ( E. w E. z ( w = <. x , y >. /\ A = <. w , z >. ) <-> E. z A = <. <. x , y >. , z >. )
21	20	2exbii 1775	. . 3 |- ( E. x E. y E. w E. z ( w = <. x , y >. /\ A = <. w , z >. ) <-> E. x E. y E. z A = <. <. x , y >. , z >. )
22	12, 13, 21	3bitr2i 288	. 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 >. )
23	1, 22	bitri 264	1 |- ( A e. ( ( _V X. _V ) X. _V ) <-> E. x E. y E. z A = <. <. x , y >. , z >. )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   <.cop 4183   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-rab 2921  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-opab 4713  df-xp 5120

This theorem is referenced by:  ssrelrel  5220  dftpos3  7370