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Theorem xp0 5254
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 4915 . . 3  |-  ( (/)  X.  A )  =  (/)
21cnveqi 5012 . 2  |-  `' (
(/)  X.  A )  =  `' (/)
3 cnvxp 5253 . 2  |-  `' (
(/)  X.  A )  =  ( A  X.  (/) )
4 cnv0 5238 . 2  |-  `' (/)  =  (/)
52, 3, 43eqtr3i 2469 1  |-  ( A  X.  (/) )  =  (/)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1369   (/)c0 3635    X. cxp 4836   `'ccnv 4837
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pr 4529
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-br 4291  df-opab 4349  df-xp 4844  df-rel 4845  df-cnv 4846
This theorem is referenced by:  xpnz  5255  xpdisj2  5258  difxp1  5261  dmxpss  5267  rnxpid  5269  xpcan  5272  unixp  5368  fconst5  5933  dfac5lem3  8293  xpcdaen  8350  fpwwe2lem13  8807  comfffval  14635  fuchom  14869  xpccofval  14990  frmdplusg  15530  mulgfval  15626  mulgfvi  15629  ga0  15814  symgplusg  15892  efgval  16212  psrplusg  17450  psrvscafval  17459  opsrle  17555  ply1plusgfvi  17695  txindislem  19204  txhaus  19218  0met  19939  zrdivrng  23917  mbfmcst  26672  0rrv  26832  dfpo2  27563  elima4  27588  isbnd3  28680
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