Theorem inxp 5140

Description: The intersection of two Cartesian products. Exercise 9 of [TakeutiZaring] p. 25. (Contributed by NM, 3-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
inxp

Proof of Theorem inxp

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

Step	Hyp	Ref	Expression
1		inopab 5138	. . 3
2		an4 824	. . . . 5
3		elin 3686	. . . . . 6
4		elin 3686	. . . . . 6
5	3, 4	anbi12i 697	. . . . 5
6	2, 5	bitr4i 252	. . . 4
7	6	opabbii 4516	. . 3
8	1, 7	eqtri 2486	. 2
9		df-xp 5010	. . 3
10		df-xp 5010	. . 3
11	9, 10	ineq12i 3697	. 2
12		df-xp 5010	. 2
13	8, 11, 12	3eqtr4i 2496	1

Syntax hints:  /\wa 369  =wceq 1395  e.wcel 1818  i^icin 3474  {copab 4509  X.cxp 5002

This theorem is referenced by:  xpindi  5141  xpindir  5142  dmxpin  5228  xpssres  5313  xpdisj1  5433  xpdisj2  5434  imainrect  5453  xpima  5454  curry1  6892  curry2  6895  fpar  6904  marypha1lem  7913  fpwwe2lem13  9041  hashxplem  12491  sscres  15192  gsumxp  17004  gsumxpOLD  17006  pjfval  18737  pjpm  18739  txbas  20068  txcls  20105  txrest  20132  trust  20732  ressuss  20766  trcfilu  20797  metreslem  20865  ressxms  21028  ressms  21029  mbfmcst  28230  0rrv  28390  xphe  37804

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-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-opab 4511  df-xp 5010  df-rel 5011