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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 464 | . . 3 ⊢ ((𝐴 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) ∧ 𝐴 = 〈𝑥, 𝑦〉)) | |
2 | 1 | 2exbii 1850 | . 2 ⊢ (∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)) ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) ∧ 𝐴 = 〈𝑥, 𝑦〉)) |
3 | elxp 5542 | . 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥∃𝑦(𝐴 = 〈𝑥, 𝑦〉 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))) | |
4 | r2ex 3262 | . 2 ⊢ (∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝐴 = 〈𝑥, 𝑦〉 ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶) ∧ 𝐴 = 〈𝑥, 𝑦〉)) | |
5 | 2, 3, 4 | 3bitr4i 306 | 1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝐴 = 〈𝑥, 𝑦〉) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1538 ∃wex 1781 ∈ wcel 2111 ∃wrex 3107 〈cop 4531 × cxp 5517 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-opab 5093 df-xp 5525 |
This theorem is referenced by: opelxp 5555 xpiundi 5586 xpiundir 5587 ssrel2 5623 reuop 6112 el2xptp 7717 f1o2ndf1 7801 xpdom2 8595 tskxpss 10183 nqereu 10340 elreal 10542 efgmnvl 18832 frgpuptinv 18889 frgpup3lem 18895 ucnima 22887 ltgseg 26390 suppovss 30443 qtophaus 31189 esum2dlem 31461 bj-mpomptALT 34534 fourierdlem42 42791 |
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