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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.
StepHypRef Expression
1 difss 3630 . . 3
2 relxp 5115 . . 3
3 relss 5095 . . 3
41, 2, 3mp2 9 . 2
5 relxp 5115 . . 3
6 relxp 5115 . . 3
7 relun 5124 . . 3
85, 6, 7mpbir2an 920 . 2
9 ianor 488 . . . . . 6
109anbi2i 694 . . . . 5
11 andi 867 . . . . 5
1210, 11bitri 249 . . . 4
13 opelxp 5034 . . . . 5
14 opelxp 5034 . . . . . 6
1514notbii 296 . . . . 5
1613, 15anbi12i 697 . . . 4
17 opelxp 5034 . . . . . 6
18 eldif 3485 . . . . . . . 8
1918anbi1i 695 . . . . . . 7
20 an32 798 . . . . . . 7
2119, 20bitri 249 . . . . . 6
2217, 21bitri 249 . . . . 5
23 eldif 3485 . . . . . . 7
2423anbi2i 694 . . . . . 6
25 opelxp 5034 . . . . . 6
26 anass 649 . . . . . 6
2724, 25, 263bitr4i 277 . . . . 5
2822, 27orbi12i 521 . . . 4
2912, 16, 283bitr4i 277 . . 3
30 eldif 3485 . . 3
31 elun 3644 . . 3
3229, 30, 313bitr4i 277 . 2
334, 8, 32eqrelriiv 5102 1
 Colors of variables: wff setvar class 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
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