Theorem elxp5 6745

Description: Membership in a Cartesian product requiring no quantifiers or dummy variables. Provides a slightly shorter version of elxp4 6744 when the double intersection does not create class existence problems (caused by int0 4300). (Contributed by NM, 1-Aug-2004.)

Assertion

Ref	Expression
elxp5

Proof of Theorem elxp5

Dummy variables are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elxp 5021	. 2
2		sneq 4039	. . . . . . . . . . . 12
3	2	rneqd 5235	. . . . . . . . . . 11
4	3	unieqd 4259	. . . . . . . . . 10
5		vex 3112	. . . . . . . . . . 11
6		vex 3112	. . . . . . . . . . 11
7	5, 6	op2nda 5498	. . . . . . . . . 10
8	4, 7	syl6req 2515	. . . . . . . . 9
9	8	pm4.71ri 633	. . . . . . . 8
10	9	anbi1i 695	. . . . . . 7
11		anass 649	. . . . . . 7
12	10, 11	bitri 249	. . . . . 6
13	12	exbii 1667	. . . . 5
14		snex 4693	. . . . . . . 8
15	14	rnex 6734	. . . . . . 7
16	15	uniex 6596	. . . . . 6
17		opeq2 4218	. . . . . . . 8
18	17	eqeq2d 2471	. . . . . . 7
19		eleq1 2529	. . . . . . . 8
20	19	anbi2d 703	. . . . . . 7
21	18, 20	anbi12d 710	. . . . . 6
22	16, 21	ceqsexv 3146	. . . . 5
23	13, 22	bitri 249	. . . 4
24		inteq 4289	. . . . . . . 8
25	24	inteqd 4291	. . . . . . 7
26	5, 16	op1stb 4722	. . . . . . 7
27	25, 26	syl6req 2515	. . . . . 6
28	27	pm4.71ri 633	. . . . 5
29	28	anbi1i 695	. . . 4
30		anass 649	. . . 4
31	23, 29, 30	3bitri 271	. . 3
32	31	exbii 1667	. 2
33		eqvisset 3117	. . . . 5
34	33	adantr 465	. . . 4
35	34	exlimiv 1722	. . 3
36		elex 3118	. . . 4
37	36	ad2antrl 727	. . 3
38		opeq1 4217	. . . . . 6
39	38	eqeq2d 2471	. . . . 5
40		eleq1 2529	. . . . . 6
41	40	anbi1d 704	. . . . 5
42	39, 41	anbi12d 710	. . . 4
43	42	ceqsexgv 3232	. . 3
44	35, 37, 43	pm5.21nii 353	. 2
45	1, 32, 44	3bitri 271	1

Syntax hints:  <->wb 184  /\wa 369  =wceq 1395  E.wex 1612  e.wcel 1818  cvv 3109  {csn 4029  <.cop 4035  U.cuni 4249  |^|cint 4286  X.cxp 5002  rancrn 5005

This theorem is referenced by:  xpnnenOLD  13943

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-8 1820  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  ax-un 6592

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-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  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-uni 4250  df-int 4287  df-br 4453  df-opab 4511  df-xp 5010  df-rel 5011  df-cnv 5012  df-dm 5014  df-rn 5015