Theorem sqxpeq0 4970

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 4747	. . 3 |- ( ( A X. A ) = (/) -> dom ( A X. A ) = dom (/) )
2		dmxpid 4768	. . 3 |- dom ( A X. A ) = A
3		dm0 4761	. . 3 |- dom (/) = (/)
4	1, 2, 3	3eqtr3g 2196	. 2 |- ( ( A X. A ) = (/) -> A = (/) )
5		xpeq0r 4969	. . 3 |- ( ( A = (/) \/ A = (/) ) -> ( A X. A ) = (/) )
6	5	orcs 725	. 2 |- ( A = (/) -> ( A X. A ) = (/) )
7	4, 6	impbii 125	1 |- ( ( A X. A ) = (/) <-> A = (/) )

Syntax hints:   <-> wb 104   = wceq 1332   (/)c0 3368   X. cxp 4545   dom cdm 4547

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-xp 4553  df-rel 4554  df-cnv 4555  df-dm 4557

This theorem is referenced by:  metn0  12586