Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  difxp Structured version   Unicode version

Theorem difxp 5421
 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 3616 . . 3
2 relxp 5100 . . 3
3 relss 5080 . . 3
41, 2, 3mp2 9 . 2
5 relxp 5100 . . 3
6 relxp 5100 . . 3
7 relun 5109 . . 3
85, 6, 7mpbir2an 920 . 2
9 ianor 488 . . . . . 6
109anbi2i 694 . . . . 5
11 andi 867 . . . . 5
1210, 11bitri 249 . . . 4
13 opelxp 5019 . . . . 5
14 opelxp 5019 . . . . . 6
1514notbii 296 . . . . 5
1613, 15anbi12i 697 . . . 4
17 opelxp 5019 . . . . . 6
18 eldif 3471 . . . . . . . 8
1918anbi1i 695 . . . . . . 7
20 an32 798 . . . . . . 7
2119, 20bitri 249 . . . . . 6
2217, 21bitri 249 . . . . 5
23 eldif 3471 . . . . . . 7
2423anbi2i 694 . . . . . 6
25 opelxp 5019 . . . . . 6
26 anass 649 . . . . . 6
2724, 25, 263bitr4i 277 . . . . 5
2822, 27orbi12i 521 . . . 4
2912, 16, 283bitr4i 277 . . 3
30 eldif 3471 . . 3
31 elun 3630 . . 3
3229, 30, 313bitr4i 277 . 2
334, 8, 32eqrelriiv 5087 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wo 368   wa 369   wceq 1383   wcel 1804   cdif 3458   cun 3459   wss 3461  cop 4020   cxp 4987   wrel 4994 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-opab 4496  df-xp 4995  df-rel 4996 This theorem is referenced by:  difxp1  5422  difxp2  5423  evlslem4OLD  18152  evlslem4  18153  txcld  20082
 Copyright terms: Public domain W3C validator