Theorem elxp7 5999

Description: Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 4962. (Contributed by NM, 19-Aug-2006.)

Assertion

Ref	Expression
elxp7	⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)))

Proof of Theorem elxp7

Step	Hyp	Ref	Expression
1		elex 2652	. 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) → 𝐴 ∈ V)
2		elex 2652	. . 3 ⊢ (𝐴 ∈ (V × V) → 𝐴 ∈ V)
3	2	adantr 272	. 2 ⊢ ((𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)) → 𝐴 ∈ V)
4		1stexg 5996	. . . . . . 7 ⊢ (𝐴 ∈ V → (1st ‘𝐴) ∈ V)
5		2ndexg 5997	. . . . . . 7 ⊢ (𝐴 ∈ V → (2nd ‘𝐴) ∈ V)
6	4, 5	jca 302	. . . . . 6 ⊢ (𝐴 ∈ V → ((1st ‘𝐴) ∈ V ∧ (2nd ‘𝐴) ∈ V))
7	6	biantrud 300	. . . . 5 ⊢ (𝐴 ∈ V → (𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ↔ (𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∧ ((1st ‘𝐴) ∈ V ∧ (2nd ‘𝐴) ∈ V))))
8		elxp6 5998	. . . . 5 ⊢ (𝐴 ∈ (V × V) ↔ (𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∧ ((1st ‘𝐴) ∈ V ∧ (2nd ‘𝐴) ∈ V)))
9	7, 8	syl6rbbr 198	. . . 4 ⊢ (𝐴 ∈ V → (𝐴 ∈ (V × V) ↔ 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩))
10	9	anbi1d 456	. . 3 ⊢ (𝐴 ∈ V → ((𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)) ↔ (𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶))))
11		elxp6 5998	. . 3 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)))
12	10, 11	syl6rbbr 198	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶))))
13	1, 3, 12	pm5.21nii 661	1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) ∈ 𝐵 ∧ (2nd ‘𝐴) ∈ 𝐶)))

Syntax hints:   ∧ wa 103   ↔ wb 104   = wceq 1299   ∈ wcel 1448  Vcvv 2641  ⟨cop 3477   × cxp 4475  ‘cfv 5059  1^st c1st 5967  2^nd c2nd 5968

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-sbc 2863  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-fo 5065  df-fv 5067  df-1st 5969  df-2nd 5970

This theorem is referenced by:  xp2  6001  unielxp  6002  1stconst  6048  2ndconst  6049  f1od2  6062