Theorem sqxpeq0 5090

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 4863	. . 3 |- ( ( A X. A ) = (/) -> dom ( A X. A ) = dom (/) )
2		dmxpid 4884	. . 3 |- dom ( A X. A ) = A
3		dm0 4877	. . 3 |- dom (/) = (/)
4	1, 2, 3	3eqtr3g 2249	. 2 |- ( ( A X. A ) = (/) -> A = (/) )
5		xpeq0r 5089	. . 3 |- ( ( A = (/) \/ A = (/) ) -> ( A X. A ) = (/) )
6	5	orcs 736	. 2 |- ( A = (/) -> ( A X. A ) = (/) )
7	4, 6	impbii 126	1 |- ( ( A X. A ) = (/) <-> A = (/) )

Syntax hints:   <-> wb 105   = wceq 1364   (/)c0 3447   X. cxp 4658   dom cdm 4660

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-br 4031  df-opab 4092  df-xp 4666  df-rel 4667  df-cnv 4668  df-dm 4670

This theorem is referenced by:  metn0  14557