elxp7 - Intuitionistic Logic Explorer

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Theorem elxp7 6130

Description: Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5085. (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 2732	. 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 ∈ V)
2		elex 2732	. . 3 ⊢ (𝐴 ∈ (V × V) → 𝐴 ∈ V)
3	2	adantr 274	. 2 ⊢ ((𝐴 ∈ (V × V) ∧ ((1^st ‘𝐴) ∈ 𝐵 ∧ (2^nd ‘𝐴) ∈ 𝐶)) → 𝐴 ∈ V)
4		elxp6 6129	. . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∧ ((1^st ‘𝐴) ∈ 𝐵 ∧ (2^nd ‘𝐴) ∈ 𝐶)))
5		elxp6 6129	. . . . 5 ⊢ (𝐴 ∈ (V × V) ↔ (𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∧ ((1^st ‘𝐴) ∈ V ∧ (2^nd ‘𝐴) ∈ V)))
6		1stexg 6127	. . . . . . 7 ⊢ (𝐴 ∈ V → (1^st ‘𝐴) ∈ V)
7		2ndexg 6128	. . . . . . 7 ⊢ (𝐴 ∈ V → (2^nd ‘𝐴) ∈ V)
8	6, 7	jca 304	. . . . . 6 ⊢ (𝐴 ∈ V → ((1^st ‘𝐴) ∈ V ∧ (2^nd ‘𝐴) ∈ V))
9	8	biantrud 302	. . . . 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 198	. . . 4 ⊢ (𝐴 ∈ V → (𝐴 ∈ (V × V) ↔ 𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩))
11	10	anbi1d 461	. . 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 198	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 ∈ (V × V) ∧ ((1^st ‘𝐴) ∈ 𝐵 ∧ (2^nd ‘𝐴) ∈ 𝐶))))
13	1, 3, 12	pm5.21nii 694	1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 ∈ (V × V) ∧ ((1^st ‘𝐴) ∈ 𝐵 ∧ (2^nd ‘𝐴) ∈ 𝐶)))

Colors of variables: wff set class

Syntax hints: ∧ wa 103 ↔ wb 104 = wceq 1342 ∈ wcel 2135 Vcvv 2721 ⟨cop 3573 × cxp 4596 ‘cfv 5182 1^st c1st 6098 2^nd c2nd 6099

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405

This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-sbc 2947 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-fo 5188 df-fv 5190 df-1st 6100 df-2nd 6101

This theorem is referenced by: xp2 6133 unielxp 6134 1stconst 6180 2ndconst 6181 f1od2 6194

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