Theorem elvvv 5064

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

Assertion

Ref	Expression
elvvv

Distinct variable group:   ,,,

Proof of Theorem elvvv

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elxp 5021	. 2
2		anass 649	. . . . 5
3		19.42vv 1777	. . . . . 6
4		ancom 450	. . . . . . 7
5	4	2exbii 1668	. . . . . 6
6		vex 3112	. . . . . . . 8
7	6	biantru 505	. . . . . . 7
8		elvv 5063	. . . . . . . 8
9	8	anbi2i 694	. . . . . . 7
10	7, 9	bitr3i 251	. . . . . 6
11	3, 5, 10	3bitr4ri 278	. . . . 5
12	2, 11	bitr3i 251	. . . 4
13	12	2exbii 1668	. . 3
14		exrot4 1853	. . . 4
15		excom 1849	. . . . . 6
16		opex 4716	. . . . . . . 8
17		opeq1 4217	. . . . . . . . 9
18	17	eqeq2d 2471	. . . . . . . 8
19	16, 18	ceqsexv 3146	. . . . . . 7
20	19	exbii 1667	. . . . . 6
21	15, 20	bitri 249	. . . . 5
22	21	2exbii 1668	. . . 4
23	14, 22	bitr3i 251	. . 3
24	13, 23	bitri 249	. 2
25	1, 24	bitri 249	1

Syntax hints:  <->wb 184  /\wa 369  =wceq 1395  E.wex 1612  e.wcel 1818  cvv 3109  <.cop 4035  X.cxp 5002

This theorem is referenced by:  ssrelrel  5108  dftpos3  6992

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-opab 4511  df-xp 5010