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Mirrors > Home > ILE Home > Th. List > xp0 | GIF version |
Description: The cross product with the empty set is empty. Part of Theorem 3.13(ii) of [Monk1] p. 37. (Contributed by NM, 12-Apr-2004.) |
Ref | Expression |
---|---|
xp0 | ⊢ (𝐴 × ∅) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0xp 4679 | . . 3 ⊢ (∅ × 𝐴) = ∅ | |
2 | 1 | cnveqi 4774 | . 2 ⊢ ◡(∅ × 𝐴) = ◡∅ |
3 | cnvxp 5017 | . 2 ⊢ ◡(∅ × 𝐴) = (𝐴 × ∅) | |
4 | cnv0 5002 | . 2 ⊢ ◡∅ = ∅ | |
5 | 2, 3, 4 | 3eqtr3i 2193 | 1 ⊢ (𝐴 × ∅) = ∅ |
Colors of variables: wff set class |
Syntax hints: = wceq 1342 ∅c0 3405 × cxp 4597 ◡ccnv 4598 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4095 ax-pow 4148 ax-pr 4182 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2724 df-dif 3114 df-un 3116 df-in 3118 df-ss 3125 df-nul 3406 df-pw 3556 df-sn 3577 df-pr 3578 df-op 3580 df-br 3978 df-opab 4039 df-xp 4605 df-rel 4606 df-cnv 4607 |
This theorem is referenced by: xpeq0r 5021 xpdisj2 5024 djuassen 7165 xpdjuen 7166 0met 12951 |
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