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Theorem sqxpeq0 5049
Description: A Cartesian square is empty iff its member is empty. (Contributed by Jim Kingdon, 21-Apr-2023.)
Assertion
Ref Expression
sqxpeq0  |-  ( ( A  X.  A )  =  (/)  <->  A  =  (/) )

Proof of Theorem sqxpeq0
StepHypRef Expression
1 dmeq 4824 . . 3  |-  ( ( A  X.  A )  =  (/)  ->  dom  ( A  X.  A )  =  dom  (/) )
2 dmxpid 4845 . . 3  |-  dom  ( A  X.  A )  =  A
3 dm0 4838 . . 3  |-  dom  (/)  =  (/)
41, 2, 33eqtr3g 2233 . 2  |-  ( ( A  X.  A )  =  (/)  ->  A  =  (/) )
5 xpeq0r 5048 . . 3  |-  ( ( A  =  (/)  \/  A  =  (/) )  ->  ( A  X.  A )  =  (/) )
65orcs 735 . 2  |-  ( A  =  (/)  ->  ( A  X.  A )  =  (/) )
74, 6impbii 126 1  |-  ( ( A  X.  A )  =  (/)  <->  A  =  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 105    = wceq 1353   (/)c0 3422    X. cxp 4622   dom cdm 4624
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4119  ax-pow 4172  ax-pr 4207
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-br 4002  df-opab 4063  df-xp 4630  df-rel 4631  df-cnv 4632  df-dm 4634
This theorem is referenced by:  metn0  13720
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