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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 4564 | . . 3 ⊢ (𝑧 ∈ (∅ × 𝐴) ↔ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴))) | |
2 | noel 3372 | . . . . . . 7 ⊢ ¬ 𝑥 ∈ ∅ | |
3 | simprl 521 | . . . . . . 7 ⊢ ((𝑧 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ ∅) | |
4 | 2, 3 | mto 652 | . . . . . 6 ⊢ ¬ (𝑧 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴)) |
5 | 4 | nex 1477 | . . . . 5 ⊢ ¬ ∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴)) |
6 | 5 | nex 1477 | . . . 4 ⊢ ¬ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴)) |
7 | noel 3372 | . . . 4 ⊢ ¬ 𝑧 ∈ ∅ | |
8 | 6, 7 | 2false 691 | . . 3 ⊢ (∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴)) ↔ 𝑧 ∈ ∅) |
9 | 1, 8 | bitri 183 | . 2 ⊢ (𝑧 ∈ (∅ × 𝐴) ↔ 𝑧 ∈ ∅) |
10 | 9 | eqriv 2137 | 1 ⊢ (∅ × 𝐴) = ∅ |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 = wceq 1332 ∃wex 1469 ∈ wcel 1481 ∅c0 3368 〈cop 3535 × cxp 4545 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-opab 3998 df-xp 4553 |
This theorem is referenced by: res0 4831 xp0 4966 xpeq0r 4969 xpdisj1 4971 xpima1 4993 xpfi 6826 exmidfodomrlemim 7074 hashxp 10604 |
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