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Mirrors > Home > ILE Home > Th. List > 0nelelxp | GIF version |
Description: A member of a cross product (ordered pair) doesn't contain the empty set. (Contributed by NM, 15-Dec-2008.) |
Ref | Expression |
---|---|
0nelelxp | ⊢ (𝐶 ∈ (𝐴 × 𝐵) → ¬ ∅ ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxp 4628 | . 2 ⊢ (𝐶 ∈ (𝐴 × 𝐵) ↔ ∃𝑥∃𝑦(𝐶 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))) | |
2 | 0nelop 4233 | . . . 4 ⊢ ¬ ∅ ∈ 〈𝑥, 𝑦〉 | |
3 | simpl 108 | . . . . 5 ⊢ ((𝐶 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝐶 = 〈𝑥, 𝑦〉) | |
4 | 3 | eleq2d 2240 | . . . 4 ⊢ ((𝐶 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (∅ ∈ 𝐶 ↔ ∅ ∈ 〈𝑥, 𝑦〉)) |
5 | 2, 4 | mtbiri 670 | . . 3 ⊢ ((𝐶 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ¬ ∅ ∈ 𝐶) |
6 | 5 | exlimivv 1889 | . 2 ⊢ (∃𝑥∃𝑦(𝐶 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ¬ ∅ ∈ 𝐶) |
7 | 1, 6 | sylbi 120 | 1 ⊢ (𝐶 ∈ (𝐴 × 𝐵) → ¬ ∅ ∈ 𝐶) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 = wceq 1348 ∃wex 1485 ∈ wcel 2141 ∅c0 3414 〈cop 3586 × cxp 4609 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-v 2732 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-opab 4051 df-xp 4617 |
This theorem is referenced by: dmsn0el 5080 |
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