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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 5708 | . 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) | |
| 2 | eqcom 2744 | . . . 4 ⊢ (〈𝑥, 𝑦〉 = 𝐴 ↔ 𝐴 = 〈𝑥, 𝑦〉) | |
| 3 | opelxp 5721 | . . . 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 1849 | . 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 1540 ∃wex 1779 ∈ wcel 2108 〈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-ral 3062 df-rex 3071 df-rab 3437 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: optocl 5780 unixp0 6303 |
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