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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 5563 | . . 3 ⊢ (𝐵 × 𝐶) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)} | |
2 | 1 | eleq2i 2906 | . 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ 𝐴 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)}) |
3 | elopab 5416 | . 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 1537 ∃wex 1780 ∈ wcel 2114 〈cop 4575 {copab 5130 × cxp 5555 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-opab 5131 df-xp 5563 |
This theorem is referenced by: elxp2 5581 0nelxp 5591 0nelelxp 5592 rabxp 5602 elxp3 5620 elvv 5628 elvvv 5629 0xp 5651 dfres3 5860 xpdifid 6027 dfco2a 6101 elsnxp 6144 tpres 6965 elxp4 7629 elxp5 7630 opabex3d 7668 opabex3rd 7669 opabex3 7670 xp1st 7723 xp2nd 7724 poxp 7824 soxp 7825 xpsnen 8603 xpcomco 8609 xpassen 8613 dfac5lem1 9551 dfac5lem4 9554 axdc4lem 9879 fsum2dlem 15127 fprod2dlem 15336 numclwwlk1lem2fo 28139 satefvfmla0 32667 elima4 33021 brcart 33395 brimg 33400 dibelval3 38285 |
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