Theorem difxp 5436

Description: Difference of Cartesian products, expressed in terms of a union of Cartesian products of differences. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 26-Jun-2014.)

Assertion

Ref	Expression
difxp

Proof of Theorem difxp

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

Step	Hyp	Ref	Expression
1		difss 3630	. . 3
2		relxp 5115	. . 3
3		relss 5095	. . 3
4	1, 2, 3	mp2 9	. 2
5		relxp 5115	. . 3
6		relxp 5115	. . 3
7		relun 5124	. . 3
8	5, 6, 7	mpbir2an 920	. 2
9		ianor 488	. . . . . 6
10	9	anbi2i 694	. . . . 5
11		andi 867	. . . . 5
12	10, 11	bitri 249	. . . 4
13		opelxp 5034	. . . . 5
14		opelxp 5034	. . . . . 6
15	14	notbii 296	. . . . 5
16	13, 15	anbi12i 697	. . . 4
17		opelxp 5034	. . . . . 6
18		eldif 3485	. . . . . . . 8
19	18	anbi1i 695	. . . . . . 7
20		an32 798	. . . . . . 7
21	19, 20	bitri 249	. . . . . 6
22	17, 21	bitri 249	. . . . 5
23		eldif 3485	. . . . . . 7
24	23	anbi2i 694	. . . . . 6
25		opelxp 5034	. . . . . 6
26		anass 649	. . . . . 6
27	24, 25, 26	3bitr4i 277	. . . . 5
28	22, 27	orbi12i 521	. . . 4
29	12, 16, 28	3bitr4i 277	. . 3
30		eldif 3485	. . 3
31		elun 3644	. . 3
32	29, 30, 31	3bitr4i 277	. 2
33	4, 8, 32	eqrelriiv 5102	1

Syntax hints:  -.wn 3  \/wo 368  /\wa 369  =wceq 1395  e.wcel 1818  \cdif 3472  u.cun 3473  C_wss 3475  <.cop 4035  X.cxp 5002  Relwrel 5009

This theorem is referenced by:  difxp1  5437  difxp2  5438  evlslem4OLD  18173  evlslem4  18174  txcld  20104

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