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Mirrors > Home > ILE Home > Th. List > 0xp | 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, 4-Jul-1994.) |
Ref | Expression |
---|---|
0xp | ⊢ (∅ × 𝐴) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxp 4380 | . . 3 ⊢ (𝑧 ∈ (∅ × 𝐴) ↔ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴))) | |
2 | noel 3255 | . . . . . . 7 ⊢ ¬ 𝑥 ∈ ∅ | |
3 | simprl 497 | . . . . . . 7 ⊢ ((𝑧 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ ∅) | |
4 | 2, 3 | mto 620 | . . . . . 6 ⊢ ¬ (𝑧 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴)) |
5 | 4 | nex 1429 | . . . . 5 ⊢ ¬ ∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴)) |
6 | 5 | nex 1429 | . . . 4 ⊢ ¬ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴)) |
7 | noel 3255 | . . . 4 ⊢ ¬ 𝑧 ∈ ∅ | |
8 | 6, 7 | 2false 649 | . . 3 ⊢ (∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴)) ↔ 𝑧 ∈ ∅) |
9 | 1, 8 | bitri 182 | . 2 ⊢ (𝑧 ∈ (∅ × 𝐴) ↔ 𝑧 ∈ ∅) |
10 | 9 | eqriv 2078 | 1 ⊢ (∅ × 𝐴) = ∅ |
Colors of variables: wff set class |
Syntax hints: ∧ wa 102 = wceq 1284 ∃wex 1421 ∈ wcel 1433 ∅c0 3251 〈cop 3401 × cxp 4361 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-opab 3840 df-xp 4369 |
This theorem is referenced by: res0 4634 xp0 4763 xpeq0r 4766 xpdisj1 4767 xpima1 4787 |
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