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Theorem xp0 5425
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 5080 . . 3  |-  ( (/)  X.  A )  =  (/)
21cnveqi 5177 . 2  |-  `' (
(/)  X.  A )  =  `' (/)
3 cnvxp 5424 . 2  |-  `' (
(/)  X.  A )  =  ( A  X.  (/) )
4 cnv0 5409 . 2  |-  `' (/)  =  (/)
52, 3, 43eqtr3i 2504 1  |-  ( A  X.  (/) )  =  (/)
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379   (/)c0 3785    X. cxp 4997   `'ccnv 4998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-rel 5006  df-cnv 5007
This theorem is referenced by:  xpnz  5426  xpdisj2  5429  difxp1  5432  dmxpss  5438  rnxpid  5440  xpcan  5443  unixp  5540  fconst5  6118  dfac5lem3  8506  xpcdaen  8563  fpwwe2lem13  9020  comfffval  14954  fuchom  15188  xpccofval  15309  frmdplusg  15854  mulgfval  15953  mulgfvi  15956  ga0  16141  symgplusg  16219  efgval  16541  psrplusg  17833  psrvscafval  17842  opsrle  17939  ply1plusgfvi  18082  txindislem  19897  txhaus  19911  0met  20632  zrdivrng  25138  mbfmcst  27898  0rrv  28058  dfpo2  28789  elima4  28814  isbnd3  29911
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