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Theorem sqxpeq0 4962
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 4739 . . 3  |-  ( ( A  X.  A )  =  (/)  ->  dom  ( A  X.  A )  =  dom  (/) )
2 dmxpid 4760 . . 3  |-  dom  ( A  X.  A )  =  A
3 dm0 4753 . . 3  |-  dom  (/)  =  (/)
41, 2, 33eqtr3g 2195 . 2  |-  ( ( A  X.  A )  =  (/)  ->  A  =  (/) )
5 xpeq0r 4961 . . 3  |-  ( ( A  =  (/)  \/  A  =  (/) )  ->  ( A  X.  A )  =  (/) )
65orcs 724 . 2  |-  ( A  =  (/)  ->  ( A  X.  A )  =  (/) )
74, 6impbii 125 1  |-  ( ( A  X.  A )  =  (/)  <->  A  =  (/) )
Colors of variables: wff set class
Syntax hints:    <-> wb 104    = wceq 1331   (/)c0 3363    X. cxp 4537   dom cdm 4539
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-in1 603  ax-in2 604  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-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-xp 4545  df-rel 4546  df-cnv 4547  df-dm 4549
This theorem is referenced by:  metn0  12557
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