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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 5708 | . 2 ⊢ (𝐶 ∈ (𝐴 × 𝐵) ↔ ∃𝑥∃𝑦(𝐶 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))) | |
| 2 | 0nelop 5501 | . . . . 5 ⊢ ¬ ∅ ∈ 〈𝑥, 𝑦〉 | |
| 3 | eleq2 2830 | . . . . 5 ⊢ (𝐶 = 〈𝑥, 𝑦〉 → (∅ ∈ 𝐶 ↔ ∅ ∈ 〈𝑥, 𝑦〉)) | |
| 4 | 2, 3 | mtbiri 327 | . . . 4 ⊢ (𝐶 = 〈𝑥, 𝑦〉 → ¬ ∅ ∈ 𝐶) |
| 5 | 4 | adantr 480 | . . 3 ⊢ ((𝐶 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ¬ ∅ ∈ 𝐶) |
| 6 | 5 | exlimivv 1932 | . 2 ⊢ (∃𝑥∃𝑦(𝐶 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ¬ ∅ ∈ 𝐶) |
| 7 | 1, 6 | sylbi 217 | 1 ⊢ (𝐶 ∈ (𝐴 × 𝐵) → ¬ ∅ ∈ 𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ∅c0 4333 〈cop 4632 × cxp 5683 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-opab 5206 df-xp 5691 |
| This theorem is referenced by: dmsn0el 6231 onxpdisj 6510 |
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