![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 5711 | . 2 ⊢ (𝐶 ∈ (𝐴 × 𝐵) ↔ ∃𝑥∃𝑦(𝐶 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))) | |
2 | 0nelop 5505 | . . . . 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 480 | . . 3 ⊢ ((𝐶 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ¬ ∅ ∈ 𝐶) |
6 | 5 | exlimivv 1929 | . 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 1536 ∃wex 1775 ∈ wcel 2105 ∅c0 4338 〈cop 4636 × cxp 5686 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-opab 5210 df-xp 5694 |
This theorem is referenced by: dmsn0el 6232 onxpdisj 6511 |
Copyright terms: Public domain | W3C validator |