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Mirrors > Home > MPE Home > Th. List > elxp | Structured version Visualization version GIF version |
Description: Membership in a Cartesian 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 5694 | . . 3 ⊢ (𝐵 × 𝐶) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)} | |
2 | 1 | eleq2i 2830 | . 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ 𝐴 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)}) |
3 | elopab 5536 | . 2 ⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)} ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) | |
4 | 2, 3 | bitri 275 | 1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1536 ∃wex 1775 ∈ wcel 2105 〈cop 4636 {copab 5209 × 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-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: elxp2 5712 0nelxp 5722 0nelelxp 5723 rabxp 5736 elxp3 5754 elvv 5762 elvvv 5763 0xp 5786 dfres3 6004 xpdifid 6189 dfco2a 6267 elsnxp 6312 tpres 7220 elxp4 7944 elxp5 7945 opabex3d 7988 opabex3rd 7989 opabex3 7990 xp1st 8044 xp2nd 8045 poxp 8151 soxp 8152 xpsnen 9093 xpcomco 9100 xpassen 9104 dfac5lem1 10160 dfac5lem4 10163 dfac5lem4OLD 10165 axdc4lem 10492 fsum2dlem 15802 fprod2dlem 16012 numclwwlk1lem2fo 30386 satefvfmla0 35402 elima4 35756 brcart 35913 brimg 35918 dibelval3 41129 |
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