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

Step	Hyp	Ref	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 (/) = (/)
4	1, 2, 3	3eqtr3g 2233	. 2 |- ( ( A X. A ) = (/) -> A = (/) )
5		xpeq0r 5048	. . 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 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