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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 5612 | . 2 ⊢ (𝐶 ∈ (𝐴 × 𝐵) ↔ ∃𝑥∃𝑦(𝐶 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))) | |
2 | 0nelop 5410 | . . . . 5 ⊢ ¬ ∅ ∈ 〈𝑥, 𝑦〉 | |
3 | eleq2 2827 | . . . . 5 ⊢ (𝐶 = 〈𝑥, 𝑦〉 → (∅ ∈ 𝐶 ↔ ∅ ∈ 〈𝑥, 𝑦〉)) | |
4 | 2, 3 | mtbiri 327 | . . . 4 ⊢ (𝐶 = 〈𝑥, 𝑦〉 → ¬ ∅ ∈ 𝐶) |
5 | 4 | adantr 481 | . . 3 ⊢ ((𝐶 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ¬ ∅ ∈ 𝐶) |
6 | 5 | exlimivv 1935 | . 2 ⊢ (∃𝑥∃𝑦(𝐶 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ¬ ∅ ∈ 𝐶) |
7 | 1, 6 | sylbi 216 | 1 ⊢ (𝐶 ∈ (𝐴 × 𝐵) → ¬ ∅ ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1539 ∃wex 1782 ∈ wcel 2106 ∅c0 4256 〈cop 4567 × cxp 5587 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-ne 2944 df-v 3434 df-dif 3890 df-un 3892 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-opab 5137 df-xp 5595 |
This theorem is referenced by: dmsn0el 6114 onxpdisj 6386 |
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