Theorem elxp3 5169

Description: Membership in a Cartesian product. (Contributed by NM, 5-Mar-1995.)

Assertion

Ref	Expression
elxp3	⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥∃𝑦(⟨𝑥, 𝑦⟩ = 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶)))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑥,𝐶,𝑦

Proof of Theorem elxp3

Step	Hyp	Ref	Expression
1		elxp 5131	. 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
2		eqcom 2629	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ = 𝐴 ↔ 𝐴 = ⟨𝑥, 𝑦⟩)
3		opelxp 5146	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶) ↔ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶))
4	2, 3	anbi12i 733	. . 3 ⊢ ((⟨𝑥, 𝑦⟩ = 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶)) ↔ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
5	4	2exbii 1775	. 2 ⊢ (∃𝑥∃𝑦(⟨𝑥, 𝑦⟩ = 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶)) ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
6	1, 5	bitr4i 267	1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥∃𝑦(⟨𝑥, 𝑦⟩ = 𝐴 ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐵 × 𝐶)))

Syntax hints:   ↔ wb 196   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ⟨cop 4183   × cxp 5112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-opab 4713  df-xp 5120

This theorem is referenced by:  optocl  5195  unixp0  5669