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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 461 | . . 3 ⊢ ((𝐴 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) ∧ 𝐴 = 〈𝑥, 𝑦〉)) | |
2 | 1 | 2exbii 1840 | . 2 ⊢ (∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) ∧ 𝐴 = 〈𝑥, 𝑦〉)) |
3 | elxp 5571 | . 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) | |
4 | r2ex 3300 | . 2 ⊢ (∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝐴 = 〈𝑥, 𝑦〉 ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) ∧ 𝐴 = 〈𝑥, 𝑦〉)) | |
5 | 2, 3, 4 | 3bitr4i 304 | 1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝐴 = 〈𝑥, 𝑦〉) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1528 ∃wex 1771 ∈ wcel 2105 ∃wrex 3136 〈cop 4563 × cxp 5546 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-opab 5120 df-xp 5554 |
This theorem is referenced by: opelxp 5584 xpiundi 5615 xpiundir 5616 ssrel2 5652 reuop 6137 el2xptp 7724 f1o2ndf1 7807 xpdom2 8600 tskxpss 10182 nqereu 10339 elreal 10541 efgmnvl 18769 frgpuptinv 18826 frgpup3lem 18832 ucnima 22817 ltgseg 26309 suppovss 30354 qtophaus 30999 esum2dlem 31250 bj-mpomptALT 34303 fourierdlem42 42311 |
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