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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 5691 | . . 3 ⊢ (𝐵 × 𝐶) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)} | |
| 2 | 1 | eleq2i 2833 | . 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ 𝐴 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)}) |
| 3 | elopab 5532 | . 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 1540 ∃wex 1779 ∈ wcel 2108 〈cop 4632 {copab 5205 × 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-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: elxp2 5709 0nelxp 5719 0nelelxp 5720 rabxp 5733 elxp3 5751 elvv 5760 elvvv 5761 0xp 5784 dfres3 6002 xpdifid 6188 dfco2a 6266 elsnxp 6311 tpres 7221 elxp4 7944 elxp5 7945 opabex3d 7990 opabex3rd 7991 opabex3 7992 xp1st 8046 xp2nd 8047 poxp 8153 soxp 8154 xpsnen 9095 xpcomco 9102 xpassen 9106 dfac5lem1 10163 dfac5lem4 10166 dfac5lem4OLD 10168 axdc4lem 10495 fsum2dlem 15806 fprod2dlem 16016 numclwwlk1lem2fo 30377 satefvfmla0 35423 elima4 35776 brcart 35933 brimg 35938 dibelval3 41149 |
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