Theorem sqxpeq0 5034

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 4811	. . 3 |- ( ( A X. A ) = (/) -> dom ( A X. A ) = dom (/) )
2		dmxpid 4832	. . 3 |- dom ( A X. A ) = A
3		dm0 4825	. . 3 |- dom (/) = (/)
4	1, 2, 3	3eqtr3g 2226	. 2 |- ( ( A X. A ) = (/) -> A = (/) )
5		xpeq0r 5033	. . 3 |- ( ( A = (/) \/ A = (/) ) -> ( A X. A ) = (/) )
6	5	orcs 730	. 2 |- ( A = (/) -> ( A X. A ) = (/) )
7	4, 6	impbii 125	1 |- ( ( A X. A ) = (/) <-> A = (/) )

Syntax hints:   <-> wb 104   = wceq 1348   (/)c0 3414   X. cxp 4609   dom cdm 4611

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-rel 4618  df-cnv 4619  df-dm 4621

This theorem is referenced by:  metn0  13172