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Mirrors > Home > ILE Home > Th. List > elxp | GIF version |
Description: Membership in a cross product. (Contributed by NM, 4-Jul-1994.) |
Ref | Expression |
---|---|
elxp | ⊢ (A ∈ (B × 𝐶) ↔ ∃x∃y(A = 〈x, y〉 ∧ (x ∈ B ∧ y ∈ 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-xp 4294 | . . 3 ⊢ (B × 𝐶) = {〈x, y〉 ∣ (x ∈ B ∧ y ∈ 𝐶)} | |
2 | 1 | eleq2i 2101 | . 2 ⊢ (A ∈ (B × 𝐶) ↔ A ∈ {〈x, y〉 ∣ (x ∈ B ∧ y ∈ 𝐶)}) |
3 | elopab 3986 | . 2 ⊢ (A ∈ {〈x, y〉 ∣ (x ∈ B ∧ y ∈ 𝐶)} ↔ ∃x∃y(A = 〈x, y〉 ∧ (x ∈ B ∧ y ∈ 𝐶))) | |
4 | 2, 3 | bitri 173 | 1 ⊢ (A ∈ (B × 𝐶) ↔ ∃x∃y(A = 〈x, y〉 ∧ (x ∈ B ∧ y ∈ 𝐶))) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 = wceq 1242 ∃wex 1378 ∈ wcel 1390 〈cop 3370 {copab 3808 × cxp 4286 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-opab 3810 df-xp 4294 |
This theorem is referenced by: elxp2 4306 0nelxp 4315 0nelelxp 4316 rabxp 4323 elxp3 4337 elvv 4345 elvvv 4346 0xp 4363 xpmlem 4687 elxp4 4751 elxp5 4752 dfco2a 4764 opabex3d 5690 opabex3 5691 xp1st 5734 xp2nd 5735 poxp 5794 xpsnen 6231 xpcomco 6236 xpassen 6240 nqnq0pi 6421 |
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