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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 4661 | . . 3 ⊢ (𝑧 ∈ (∅ × 𝐴) ↔ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴))) | |
2 | noel 3441 | . . . . . . 7 ⊢ ¬ 𝑥 ∈ ∅ | |
3 | simprl 529 | . . . . . . 7 ⊢ ((𝑧 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ ∅) | |
4 | 2, 3 | mto 663 | . . . . . 6 ⊢ ¬ (𝑧 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴)) |
5 | 4 | nex 1511 | . . . . 5 ⊢ ¬ ∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴)) |
6 | 5 | nex 1511 | . . . 4 ⊢ ¬ ∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴)) |
7 | noel 3441 | . . . 4 ⊢ ¬ 𝑧 ∈ ∅ | |
8 | 6, 7 | 2false 702 | . . 3 ⊢ (∃𝑥∃𝑦(𝑧 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ ∅ ∧ 𝑦 ∈ 𝐴)) ↔ 𝑧 ∈ ∅) |
9 | 1, 8 | bitri 184 | . 2 ⊢ (𝑧 ∈ (∅ × 𝐴) ↔ 𝑧 ∈ ∅) |
10 | 9 | eqriv 2186 | 1 ⊢ (∅ × 𝐴) = ∅ |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 = wceq 1364 ∃wex 1503 ∈ wcel 2160 ∅c0 3437 〈cop 3610 × cxp 4642 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-opab 4080 df-xp 4650 |
This theorem is referenced by: res0 4929 xp0 5066 xpeq0r 5069 xpdisj1 5071 xpima1 5093 xpfi 6957 exmidfodomrlemim 7229 hashxp 10837 |
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