elxp7 - Intuitionistic Logic Explorer

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Theorem elxp7 6246

Description: Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5167. (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		elex 2782	. 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 ∈ V)
2		elex 2782	. . 3 ⊢ (𝐴 ∈ (V × V) → 𝐴 ∈ V)
3	2	adantr 276	. 2 ⊢ ((𝐴 ∈ (V × V) ∧ ((1^st ‘𝐴) ∈ 𝐵 ∧ (2^nd ‘𝐴) ∈ 𝐶)) → 𝐴 ∈ V)
4		elxp6 6245	. . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∧ ((1^st ‘𝐴) ∈ 𝐵 ∧ (2^nd ‘𝐴) ∈ 𝐶)))
5		elxp6 6245	. . . . 5 ⊢ (𝐴 ∈ (V × V) ↔ (𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∧ ((1^st ‘𝐴) ∈ V ∧ (2^nd ‘𝐴) ∈ V)))
6		1stexg 6243	. . . . . . 7 ⊢ (𝐴 ∈ V → (1^st ‘𝐴) ∈ V)
7		2ndexg 6244	. . . . . . 7 ⊢ (𝐴 ∈ V → (2^nd ‘𝐴) ∈ V)
8	6, 7	jca 306	. . . . . 6 ⊢ (𝐴 ∈ V → ((1^st ‘𝐴) ∈ V ∧ (2^nd ‘𝐴) ∈ V))
9	8	biantrud 304	. . . . 5 ⊢ (𝐴 ∈ V → (𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ↔ (𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∧ ((1^st ‘𝐴) ∈ V ∧ (2^nd ‘𝐴) ∈ V))))
10	5, 9	bitr4id 199	. . . 4 ⊢ (𝐴 ∈ V → (𝐴 ∈ (V × V) ↔ 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩))
11	10	anbi1d 465	. . 3 ⊢ (𝐴 ∈ V → ((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))))
12	4, 11	bitr4id 199	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 ∈ (V × V) ∧ ((1^st ‘𝐴) ∈ 𝐵 ∧ (2^nd ‘𝐴) ∈ 𝐶))))
13	1, 3, 12	pm5.21nii 705	1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 ∈ (V × V) ∧ ((1^st ‘𝐴) ∈ 𝐵 ∧ (2^nd ‘𝐴) ∈ 𝐶)))

Colors of variables: wff set class

Syntax hints: ∧ wa 104 ↔ wb 105 = wceq 1372 ∈ wcel 2175 Vcvv 2771 ⟨cop 3635 × cxp 4671 ‘cfv 5268 1^st c1st 6214 2^nd c2nd 6215

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1469 ax-7 1470 ax-gen 1471 ax-ie1 1515 ax-ie2 1516 ax-8 1526 ax-10 1527 ax-11 1528 ax-i12 1529 ax-bndl 1531 ax-4 1532 ax-17 1548 ax-i9 1552 ax-ial 1556 ax-i5r 1557 ax-13 2177 ax-14 2178 ax-ext 2186 ax-sep 4161 ax-pow 4217 ax-pr 4252 ax-un 4478

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1375 df-nf 1483 df-sb 1785 df-eu 2056 df-mo 2057 df-clab 2191 df-cleq 2197 df-clel 2200 df-nfc 2336 df-ral 2488 df-rex 2489 df-v 2773 df-sbc 2998 df-un 3169 df-in 3171 df-ss 3178 df-pw 3617 df-sn 3638 df-pr 3639 df-op 3641 df-uni 3850 df-br 4044 df-opab 4105 df-mpt 4106 df-id 4338 df-xp 4679 df-rel 4680 df-cnv 4681 df-co 4682 df-dm 4683 df-rn 4684 df-iota 5229 df-fun 5270 df-fn 5271 df-f 5272 df-fo 5274 df-fv 5276 df-1st 6216 df-2nd 6217

This theorem is referenced by: xp2 6249 unielxp 6250 1stconst 6297 2ndconst 6298 f1od2 6311

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