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Theorem elxp7 7968

Description: Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp4 7864. (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 7967	. 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∧ ((1^st ‘𝐴) ∈ 𝐵 ∧ (2^nd ‘𝐴) ∈ 𝐶)))
2		fvex 6847	. . . . 5 ⊢ (1^st ‘𝐴) ∈ V
3		fvex 6847	. . . . 5 ⊢ (2^nd ‘𝐴) ∈ V
4	2, 3	pm3.2i 470	. . . 4 ⊢ ((1^st ‘𝐴) ∈ V ∧ (2^nd ‘𝐴) ∈ V)
5		elxp6 7967	. . . 4 ⊢ (𝐴 ∈ (V × V) ↔ (𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∧ ((1^st ‘𝐴) ∈ V ∧ (2^nd ‘𝐴) ∈ V)))
6	4, 5	mpbiran2 710	. . 3 ⊢ (𝐴 ∈ (V × V) ↔ 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
7	6	anbi1i 624	. 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 206 ∧ wa 395 = wceq 1541 ∈ wcel 2113 Vcvv 3440 ⟨cop 4586 × cxp 5622 ‘cfv 6492 1^st c1st 7931 2^nd c2nd 7932

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 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-sep 5241 ax-nul 5251 ax-pr 5377 ax-un 7680

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3061 df-rab 3400 df-v 3442 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-nul 4286 df-if 4480 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-br 5099 df-opab 5161 df-mpt 5180 df-id 5519 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-iota 6448 df-fun 6494 df-fv 6500 df-1st 7933 df-2nd 7934

This theorem is referenced by: xp2 7970 unielxp 7971 1stconst 8042 2ndconst 8043 fparlem1 8054 fparlem2 8055 infxpenlem 9923 1stpreimas 32785 1stpreima 32786 2ndpreima 32787 f1od2 32798 xpinpreima2 34064 tpr2rico 34069 sxbrsigalem0 34428 dya2iocnrect 34438 elxp8 37576 pellex 43077 elpglem3 49958

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