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Mirrors > Home > MPE Home > Th. List > elxp3 | Structured version Visualization version GIF version |
Description: Membership in a Cartesian product. (Contributed by NM, 5-Mar-1995.) |
Ref | Expression |
---|---|
elxp3 | ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥∃𝑦(〈𝑥, 𝑦〉 = 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxp 5711 | . 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) | |
2 | eqcom 2741 | . . . 4 ⊢ (〈𝑥, 𝑦〉 = 𝐴 ↔ 𝐴 = 〈𝑥, 𝑦〉) | |
3 | opelxp 5724 | . . . 4 ⊢ (〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) | |
4 | 2, 3 | anbi12i 628 | . . 3 ⊢ ((〈𝑥, 𝑦〉 = 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐶)) ↔ (𝐴 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) |
5 | 4 | 2exbii 1845 | . 2 ⊢ (∃𝑥∃𝑦(〈𝑥, 𝑦〉 = 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐶)) ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) |
6 | 1, 5 | bitr4i 278 | 1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥∃𝑦(〈𝑥, 𝑦〉 = 𝐴 ∧ 〈𝑥, 𝑦〉 ∈ (𝐵 × 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1536 ∃wex 1775 ∈ wcel 2105 〈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-ral 3059 df-rex 3068 df-rab 3433 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: optocl 5782 unixp0 6304 |
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