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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 5555 | . . 3 ⊢ (𝐵 × 𝐶) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)} | |
2 | 1 | eleq2i 2904 | . 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ 𝐴 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)}) |
3 | elopab 5406 | . 2 ⊢ (𝐴 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)} ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) | |
4 | 2, 3 | bitri 277 | 1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1533 ∃wex 1776 ∈ wcel 2110 〈cop 4566 {copab 5120 × cxp 5547 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-opab 5121 df-xp 5555 |
This theorem is referenced by: elxp2 5573 0nelxp 5583 0nelelxp 5584 rabxp 5594 elxp3 5612 elvv 5620 elvvv 5621 0xp 5643 dfres3 5852 xpdifid 6019 dfco2a 6093 elsnxp 6136 tpres 6957 elxp4 7621 elxp5 7622 opabex3d 7660 opabex3rd 7661 opabex3 7662 xp1st 7715 xp2nd 7716 poxp 7816 soxp 7817 xpsnen 8595 xpcomco 8601 xpassen 8605 dfac5lem1 9543 dfac5lem4 9546 axdc4lem 9871 fsum2dlem 15119 fprod2dlem 15328 numclwwlk1lem2fo 28131 satefvfmla0 32660 elima4 33014 brcart 33388 brimg 33393 dibelval3 38277 |
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