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| Mirrors > Home > MPE Home > Th. List > elxp2 | Structured version Visualization version GIF version | ||
| Description: Membership in a Cartesian product. (Contributed by NM, 23-Feb-2004.) (Proof shortened by JJ, 13-Aug-2021.) |
| Ref | Expression |
|---|---|
| elxp2 | ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝐴 = 〈𝑥, 𝑦〉) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ancom 460 | . . 3 ⊢ ((𝐴 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) ∧ 𝐴 = 〈𝑥, 𝑦〉)) | |
| 2 | 1 | 2exbii 1849 | . 2 ⊢ (∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) ∧ 𝐴 = 〈𝑥, 𝑦〉)) |
| 3 | elxp 5708 | . 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) | |
| 4 | r2ex 3196 | . 2 ⊢ (∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝐴 = 〈𝑥, 𝑦〉 ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) ∧ 𝐴 = 〈𝑥, 𝑦〉)) | |
| 5 | 2, 3, 4 | 3bitr4i 303 | 1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝐴 = 〈𝑥, 𝑦〉) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ∃wrex 3070 〈cop 4632 × 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-ral 3062 df-rex 3071 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: opelxp 5721 xpiundi 5756 xpiundir 5757 ssrel2 5795 reuop 6313 el2xptp 8060 f1o2ndf1 8147 frpoins3xpg 8165 poxp2 8168 xpord2pred 8170 sexp2 8171 xpdom2 9107 tskxpss 10812 nqereu 10969 elreal 11171 xpsmnd0 18791 efgmnvl 19732 frgpuptinv 19789 frgpup3lem 19795 xpsring1d 20330 pzriprnglem3 21494 pzriprnglem8 21499 pzriprnglem10 21501 ucnima 24290 ltgseg 28604 suppovss 32690 elrlocbasi 33270 qtophaus 33835 esum2dlem 34093 bj-mpomptALT 37120 fourierdlem42 46164 gpgvtxel 48005 |
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