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Theorem elxp7 8003

Description: Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp4 7906. (Contributed by NM, 19-Aug-2006.)

**Assertion**
Ref	Expression
elxp7	⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 ∈ (V × V) ∧ ((1^st ‘𝐴) ∈ 𝐵 ∧ (2^nd ‘𝐴) ∈ 𝐶)))

**Proof of Theorem elxp7**
Step	Hyp	Ref	Expression
1		elxp6 8002	. 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∧ ((1^st ‘𝐴) ∈ 𝐵 ∧ (2^nd ‘𝐴) ∈ 𝐶)))
2		fvex 6894	. . . . 5 ⊢ (1^st ‘𝐴) ∈ V
3		fvex 6894	. . . . 5 ⊢ (2^nd ‘𝐴) ∈ V
4	2, 3	pm3.2i 470	. . . 4 ⊢ ((1^st ‘𝐴) ∈ V ∧ (2^nd ‘𝐴) ∈ V)
5		elxp6 8002	. . . 4 ⊢ (𝐴 ∈ (V × V) ↔ (𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∧ ((1^st ‘𝐴) ∈ V ∧ (2^nd ‘𝐴) ∈ V)))
6	4, 5	mpbiran2 707	. . 3 ⊢ (𝐴 ∈ (V × V) ↔ 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
7	6	anbi1i 623	. 2 ⊢ ((𝐴 ∈ (V × V) ∧ ((1^st ‘𝐴) ∈ 𝐵 ∧ (2^nd ‘𝐴) ∈ 𝐶)) ↔ (𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∧ ((1^st ‘𝐴) ∈ 𝐵 ∧ (2^nd ‘𝐴) ∈ 𝐶)))
8	1, 7	bitr4i 278	1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 ∈ (V × V) ∧ ((1^st ‘𝐴) ∈ 𝐵 ∧ (2^nd ‘𝐴) ∈ 𝐶)))

Colors of variables: wff setvar class

Syntax hints: ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 Vcvv 3466 ⟨cop 4626 × cxp 5664 ‘cfv 6533 1^st c1st 7966 2^nd c2nd 7967

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 ax-un 7718

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-iota 6485 df-fun 6535 df-fv 6541 df-1st 7968 df-2nd 7969

This theorem is referenced by: xp2 8005 unielxp 8006 1stconst 8080 2ndconst 8081 fparlem1 8092 fparlem2 8093 infxpenlem 10004 1stpreimas 32396 1stpreima 32397 2ndpreima 32398 f1od2 32415 xpinpreima2 33376 tpr2rico 33381 sxbrsigalem0 33759 dya2iocnrect 33769 elxp8 36742 pellex 42062 elpglem3 47946

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