Theorem sqxpeq0 5044

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

Step	Hyp	Ref	Expression
1		dmeq 4820	. . 3 |- ( ( A X. A ) = (/) -> dom ( A X. A ) = dom (/) )
2		dmxpid 4841	. . 3 |- dom ( A X. A ) = A
3		dm0 4834	. . 3 |- dom (/) = (/)
4	1, 2, 3	3eqtr3g 2231	. 2 |- ( ( A X. A ) = (/) -> A = (/) )
5		xpeq0r 5043	. . 3 |- ( ( A = (/) \/ A = (/) ) -> ( A X. A ) = (/) )
6	5	orcs 735	. 2 |- ( A = (/) -> ( A X. A ) = (/) )
7	4, 6	impbii 126	1 |- ( ( A X. A ) = (/) <-> A = (/) )

Syntax hints:   <-> wb 105   = wceq 1353   (/)c0 3420   X. cxp 4618   dom cdm 4620

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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-nul 3421  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-br 3999  df-opab 4060  df-xp 4626  df-rel 4627  df-cnv 4628  df-dm 4630

This theorem is referenced by:  metn0  13458