Theorem ixpin 7514

Description: The intersection of two infinite Cartesian products. (Contributed by Mario Carneiro, 3-Feb-2015.)

Assertion

Ref	Expression
ixpin

Distinct variable group:   ,

Proof of Theorem ixpin

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		anandi 828	. . . 4
2		elin 3686	. . . . . . 7
3	2	ralbii 2888	. . . . . 6
4		r19.26 2984	. . . . . 6
5	3, 4	bitri 249	. . . . 5
6	5	anbi2i 694	. . . 4
7		vex 3112	. . . . . 6
8	7	elixp 7496	. . . . 5
9	7	elixp 7496	. . . . 5
10	8, 9	anbi12i 697	. . . 4
11	1, 6, 10	3bitr4i 277	. . 3
12	7	elixp 7496	. . 3
13		elin 3686	. . 3
14	11, 12, 13	3bitr4i 277	. 2
15	14	eqriv 2453	1

Syntax hints:  /\wa 369  =wceq 1395  e.wcel 1818  A.wral 2807  i^icin 3474  Fnwfn 5588  `cfv 5593  X_cixp 7489

This theorem is referenced by:  ptbasin  20078  ptclsg  20116  ptrest  30048

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-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435

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-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-br 4453  df-opab 4511  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-iota 5556  df-fun 5595  df-fn 5596  df-fv 5601  df-ixp 7490