elxp7 - Intuitionistic Logic Explorer

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Theorem elxp7 6332

Description: Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 5224. (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 2814	. 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 ∈ V)
2		elex 2814	. . 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 6331	. . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∧ ((1^st ‘𝐴) ∈ 𝐵 ∧ (2^nd ‘𝐴) ∈ 𝐶)))
5		elxp6 6331	. . . . 5 ⊢ (𝐴 ∈ (V × V) ↔ (𝐴 = ⟨(1^st ‘𝐴), (2^nd ‘𝐴)⟩ ∧ ((1^st ‘𝐴) ∈ V ∧ (2^nd ‘𝐴) ∈ V)))
6		1stexg 6329	. . . . . . 7 ⊢ (𝐴 ∈ V → (1^st ‘𝐴) ∈ V)
7		2ndexg 6330	. . . . . . 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 711	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 1397 ∈ wcel 2202 Vcvv 2802 ⟨cop 3672 × cxp 4723 ‘cfv 5326 1^st c1st 6300 2^nd c2nd 6301

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 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-sep 4207 ax-pow 4264 ax-pr 4299 ax-un 4530

This theorem depends on definitions: df-bi 117 df-3an 1006 df-tru 1400 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ral 2515 df-rex 2516 df-v 2804 df-sbc 3032 df-un 3204 df-in 3206 df-ss 3213 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-br 4089 df-opab 4151 df-mpt 4152 df-id 4390 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-fo 5332 df-fv 5334 df-1st 6302 df-2nd 6303

This theorem is referenced by: xp2 6335 unielxp 6336 1stconst 6385 2ndconst 6386 f1od2 6399

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