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Theorem elxpi 4555
 Description: Membership in a cross product. Uses fewer axioms than elxp 4556. (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 2146 . . . . . 6
21anbi1d 460 . . . . 5
322exbidv 1840 . . . 4
43elabg 2830 . . 3
54ibi 175 . 2
6 df-xp 4545 . . 3
7 df-opab 3990 . . 3
86, 7eqtri 2160 . 2
95, 8eleq2s 2234 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103   wceq 1331  wex 1468   wcel 1480  cab 2125  cop 3530  copab 3988   cxp 4537 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-opab 3990  df-xp 4545 This theorem is referenced by:  xpsspw  4651  dmaddpqlem  7185  nqpi  7186  enq0ref  7241  nqnq0  7249  nq0nn  7250  cnm  7640  axaddcl  7672  axmulcl  7674
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