ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sqxpeq0 GIF version

Theorem sqxpeq0 4920
Description: A Cartesian square is empty iff its member is empty. (Contributed by Jim Kingdon, 21-Apr-2023.)
Assertion
Ref Expression
sqxpeq0 ((𝐴 × 𝐴) = ∅ ↔ 𝐴 = ∅)

Proof of Theorem sqxpeq0
StepHypRef Expression
1 dmeq 4699 . . 3 ((𝐴 × 𝐴) = ∅ → dom (𝐴 × 𝐴) = dom ∅)
2 dmxpid 4720 . . 3 dom (𝐴 × 𝐴) = 𝐴
3 dm0 4713 . . 3 dom ∅ = ∅
41, 2, 33eqtr3g 2170 . 2 ((𝐴 × 𝐴) = ∅ → 𝐴 = ∅)
5 xpeq0r 4919 . . 3 ((𝐴 = ∅ ∨ 𝐴 = ∅) → (𝐴 × 𝐴) = ∅)
65orcs 707 . 2 (𝐴 = ∅ → (𝐴 × 𝐴) = ∅)
74, 6impbii 125 1 ((𝐴 × 𝐴) = ∅ ↔ 𝐴 = ∅)
Colors of variables: wff set class
Syntax hints:  wb 104   = wceq 1314  c0 3329   × cxp 4497  dom cdm 4499
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-v 2659  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-br 3896  df-opab 3950  df-xp 4505  df-rel 4506  df-cnv 4507  df-dm 4509
This theorem is referenced by:  metn0  12367
  Copyright terms: Public domain W3C validator