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Mirrors > Home > MPE Home > Th. List > 0nelelxp | Structured version Visualization version GIF version |
Description: A member of a Cartesian 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 5549 | . 2 ⊢ (𝐶 ∈ (𝐴 × 𝐵) ↔ ∃𝑥∃𝑦(𝐶 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))) | |
2 | 0nelop 5354 | . . . . 5 ⊢ ¬ ∅ ∈ 〈𝑥, 𝑦〉 | |
3 | eleq2 2821 | . . . . 5 ⊢ (𝐶 = 〈𝑥, 𝑦〉 → (∅ ∈ 𝐶 ↔ ∅ ∈ 〈𝑥, 𝑦〉)) | |
4 | 2, 3 | mtbiri 330 | . . . 4 ⊢ (𝐶 = 〈𝑥, 𝑦〉 → ¬ ∅ ∈ 𝐶) |
5 | 4 | adantr 484 | . . 3 ⊢ ((𝐶 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ¬ ∅ ∈ 𝐶) |
6 | 5 | exlimivv 1938 | . 2 ⊢ (∃𝑥∃𝑦(𝐶 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ¬ ∅ ∈ 𝐶) |
7 | 1, 6 | sylbi 220 | 1 ⊢ (𝐶 ∈ (𝐴 × 𝐵) → ¬ ∅ ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1542 ∃wex 1786 ∈ wcel 2113 ∅c0 4212 〈cop 4523 × cxp 5524 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-ext 2710 ax-sep 5168 ax-nul 5175 ax-pr 5297 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-sb 2074 df-clab 2717 df-cleq 2730 df-clel 2811 df-ne 2935 df-v 3400 df-dif 3847 df-un 3849 df-nul 4213 df-if 4416 df-sn 4518 df-pr 4520 df-op 4524 df-opab 5094 df-xp 5532 |
This theorem is referenced by: dmsn0el 6044 onxpdisj 6293 |
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