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Theorem xp0 5410
Description: The Cartesian product with the empty set is empty. Part of Theorem 3.13(ii) of [Monk1] p. 37. (Contributed by NM, 12-Apr-2004.)
Assertion
Ref Expression
xp0  |-  ( A  X.  (/) )  =  (/)

Proof of Theorem xp0
StepHypRef Expression
1 0xp 5069 . . 3  |-  ( (/)  X.  A )  =  (/)
21cnveqi 5166 . 2  |-  `' (
(/)  X.  A )  =  `' (/)
3 cnvxp 5409 . 2  |-  `' (
(/)  X.  A )  =  ( A  X.  (/) )
4 cnv0 5394 . 2  |-  `' (/)  =  (/)
52, 3, 43eqtr3i 2491 1  |-  ( A  X.  (/) )  =  (/)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1398   (/)c0 3783    X. cxp 4986   `'ccnv 4987
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-xp 4994  df-rel 4995  df-cnv 4996
This theorem is referenced by:  xpnz  5411  xpdisj2  5414  difxp1  5417  dmxpss  5423  rnxpid  5425  xpcan  5428  unixp  5523  fconst5  6105  dfac5lem3  8497  xpcdaen  8554  fpwwe2lem13  9009  comfffval  15186  0ssc  15325  fuchom  15449  xpccofval  15650  frmdplusg  16221  mulgfval  16342  mulgfvi  16345  ga0  16535  symgplusg  16613  efgval  16934  psrplusg  18229  psrvscafval  18238  opsrle  18335  ply1plusgfvi  18478  txindislem  20300  txhaus  20314  0met  21035  zrdivrng  25632  aciunf1  27730  mbfmcst  28467  0rrv  28654  mexval  29126  mdvval  29128  mpstval  29159  dfpo2  29425  elima4  29449  isbnd3  30520
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