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Mirrors > Home > ILE Home > Th. List > sqxpeq0 | GIF version |
Description: A Cartesian square is empty iff its member is empty. (Contributed by Jim Kingdon, 21-Apr-2023.) |
Ref | Expression |
---|---|
sqxpeq0 | ⊢ ((𝐴 × 𝐴) = ∅ ↔ 𝐴 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmeq 4699 | . . 3 ⊢ ((𝐴 × 𝐴) = ∅ → dom (𝐴 × 𝐴) = dom ∅) | |
2 | dmxpid 4720 | . . 3 ⊢ dom (𝐴 × 𝐴) = 𝐴 | |
3 | dm0 4713 | . . 3 ⊢ dom ∅ = ∅ | |
4 | 1, 2, 3 | 3eqtr3g 2170 | . 2 ⊢ ((𝐴 × 𝐴) = ∅ → 𝐴 = ∅) |
5 | xpeq0r 4919 | . . 3 ⊢ ((𝐴 = ∅ ∨ 𝐴 = ∅) → (𝐴 × 𝐴) = ∅) | |
6 | 5 | orcs 707 | . 2 ⊢ (𝐴 = ∅ → (𝐴 × 𝐴) = ∅) |
7 | 4, 6 | impbii 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 |
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