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Theorem elxp7 7978

Description: Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp4 7874. (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 7977	. 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∧ ((1^st ‘𝐴) ∈ 𝐵 ∧ (2^nd ‘𝐴) ∈ 𝐶)))
2		fvex 6855	. . . . 5 ⊢ (1^st ‘𝐴) ∈ V
3		fvex 6855	. . . . 5 ⊢ (2^nd ‘𝐴) ∈ V
4	2, 3	pm3.2i 470	. . . 4 ⊢ ((1^st ‘𝐴) ∈ V ∧ (2^nd ‘𝐴) ∈ V)
5		elxp6 7977	. . . 4 ⊢ (𝐴 ∈ (V × V) ↔ (𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∧ ((1^st ‘𝐴) ∈ V ∧ (2^nd ‘𝐴) ∈ V)))
6	4, 5	mpbiran2 711	. . 3 ⊢ (𝐴 ∈ (V × V) ↔ 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩)
7	6	anbi1i 625	. 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 1542 ∈ wcel 2114 Vcvv 3442 ⟨cop 4588 × cxp 5630 ‘cfv 6500 1^st c1st 7941 2^nd c2nd 7942

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5243 ax-nul 5253 ax-pr 5379 ax-un 7690

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-rab 3402 df-v 3444 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-nul 4288 df-if 4482 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-br 5101 df-opab 5163 df-mpt 5182 df-id 5527 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-iota 6456 df-fun 6502 df-fv 6508 df-1st 7943 df-2nd 7944

This theorem is referenced by: xp2 7980 unielxp 7981 1stconst 8052 2ndconst 8053 fparlem1 8064 fparlem2 8065 infxpenlem 9935 1stpreimas 32796 1stpreima 32797 2ndpreima 32798 f1od2 32809 xpinpreima2 34085 tpr2rico 34090 sxbrsigalem0 34449 dya2iocnrect 34459 elxp8 37626 pellex 43192 elpglem3 50072

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