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Theorem elxpi 4361
 Description: Membership in a cross product. Uses fewer axioms than elxp 4362. (Contributed by NM, 4-Jul-1994.)
Assertion
Ref Expression
elxpi
Distinct variable groups:   ,,   ,,   ,,

Proof of Theorem elxpi
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2046 . . . . . 6
21anbi1d 438 . . . . 5
322exbidv 1748 . . . 4
43elabg 2688 . . 3
54ibi 165 . 2
6 df-xp 4351 . . 3
7 df-opab 3819 . . 3
86, 7eqtri 2060 . 2
95, 8eleq2s 2132 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 97   wceq 1243  wex 1381   wcel 1393  cab 2026  cop 3378  copab 3817   cxp 4343 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-opab 3819  df-xp 4351 This theorem is referenced by:  xpsspw  4450  dmaddpqlem  6475  nqpi  6476  enq0ref  6531  nqnq0  6539  nq0nn  6540  axaddcl  6940  axmulcl  6942
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