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Theorem difxp 6409
 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.
StepHypRef Expression
1 difss 3460 . . 3
2 relxp 5012 . . 3
3 relss 4992 . . 3
41, 2, 3mp2 9 . 2
5 relxp 5012 . . 3
6 relxp 5012 . . 3
7 relun 5020 . . 3
85, 6, 7mpbir2an 888 . 2
9 ianor 476 . . . . . 6
109anbi2i 677 . . . . 5
11 andi 839 . . . . 5
1210, 11bitri 242 . . . 4
13 opelxp 4937 . . . . 5
14 opelxp 4937 . . . . . 6
1514notbii 289 . . . . 5
1613, 15anbi12i 680 . . . 4
17 opelxp 4937 . . . . . 6
18 eldif 3316 . . . . . . . 8
1918anbi1i 678 . . . . . . 7
20 an32 775 . . . . . . 7
2119, 20bitri 242 . . . . . 6
2217, 21bitri 242 . . . . 5
23 eldif 3316 . . . . . . 7
2423anbi2i 677 . . . . . 6
25 opelxp 4937 . . . . . 6
26 anass 632 . . . . . 6
2724, 25, 263bitr4i 270 . . . . 5
2822, 27orbi12i 509 . . . 4
2912, 16, 283bitr4i 270 . . 3
30 eldif 3316 . . 3
31 elun 3474 . . 3
3229, 30, 313bitr4i 270 . 2
334, 8, 32eqrelriiv 4999 1
 Colors of variables: wff set class Syntax hints:   wn 3   wo 359   wa 360   wceq 1653   wcel 1727   cdif 3303   cun 3304   wss 3306  cop 3841   cxp 4905   wrel 4912 This theorem is referenced by:  difxp1  6410  difxp2  6411  evlslem4  16595  txcld  17666 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pr 4432 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-opab 4292  df-xp 4913  df-rel 4914
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