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Mirrors > Home > ILE Home > Th. List > elxp2 | GIF version |
Description: Membership in a cross product. (Contributed by NM, 23-Feb-2004.) |
Ref | Expression |
---|---|
elxp2 | ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝐴 = 〈𝑥, 𝑦〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rex 2354 | . . . 4 ⊢ (∃𝑦 ∈ 𝐶 (𝑥 ∈ 𝐵 ∧ 𝐴 = 〈𝑥, 𝑦〉) ↔ ∃𝑦(𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐵 ∧ 𝐴 = 〈𝑥, 𝑦〉))) | |
2 | r19.42v 2511 | . . . 4 ⊢ (∃𝑦 ∈ 𝐶 (𝑥 ∈ 𝐵 ∧ 𝐴 = 〈𝑥, 𝑦〉) ↔ (𝑥 ∈ 𝐵 ∧ ∃𝑦 ∈ 𝐶 𝐴 = 〈𝑥, 𝑦〉)) | |
3 | an13 527 | . . . . 5 ⊢ ((𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐵 ∧ 𝐴 = 〈𝑥, 𝑦〉)) ↔ (𝐴 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) | |
4 | 3 | exbii 1536 | . . . 4 ⊢ (∃𝑦(𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐵 ∧ 𝐴 = 〈𝑥, 𝑦〉)) ↔ ∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) |
5 | 1, 2, 4 | 3bitr3i 208 | . . 3 ⊢ ((𝑥 ∈ 𝐵 ∧ ∃𝑦 ∈ 𝐶 𝐴 = 〈𝑥, 𝑦〉) ↔ ∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) |
6 | 5 | exbii 1536 | . 2 ⊢ (∃𝑥(𝑥 ∈ 𝐵 ∧ ∃𝑦 ∈ 𝐶 𝐴 = 〈𝑥, 𝑦〉) ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) |
7 | df-rex 2354 | . 2 ⊢ (∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝐴 = 〈𝑥, 𝑦〉 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ ∃𝑦 ∈ 𝐶 𝐴 = 〈𝑥, 𝑦〉)) | |
8 | elxp 4380 | . 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) | |
9 | 6, 7, 8 | 3bitr4ri 211 | 1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝐴 = 〈𝑥, 𝑦〉) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 102 ↔ wb 103 = wceq 1284 ∃wex 1421 ∈ wcel 1433 ∃wrex 2349 〈cop 3401 × cxp 4361 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-rex 2354 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-opab 3840 df-xp 4369 |
This theorem is referenced by: opelxp 4392 xpiundi 4416 xpiundir 4417 ssrel2 4448 f1o2ndf1 5869 xpdom2 6328 elreal 6997 |
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