Theorem elxp2OLD 5133

Description: Obsolete proof of elxp2 5132 as of 13-Aug-2021. (Contributed by NM, 23-Feb-2004.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
elxp2OLD	⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝐴 = ⟨𝑥, 𝑦⟩)

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

Proof of Theorem elxp2OLD

Step	Hyp	Ref	Expression
1		df-rex 2918	. . . 4 ⊢ (∃𝑦 ∈ 𝐶 (𝑥 ∈ 𝐵 ∧ 𝐴 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑦(𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐵 ∧ 𝐴 = ⟨𝑥, 𝑦⟩)))
2		r19.42v 3092	. . . 4 ⊢ (∃𝑦 ∈ 𝐶 (𝑥 ∈ 𝐵 ∧ 𝐴 = ⟨𝑥, 𝑦⟩) ↔ (𝑥 ∈ 𝐵 ∧ ∃𝑦 ∈ 𝐶 𝐴 = ⟨𝑥, 𝑦⟩))
3		an13 840	. . . . 5 ⊢ ((𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐵 ∧ 𝐴 = ⟨𝑥, 𝑦⟩)) ↔ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
4	3	exbii 1774	. . . 4 ⊢ (∃𝑦(𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐵 ∧ 𝐴 = ⟨𝑥, 𝑦⟩)) ↔ ∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
5	1, 2, 4	3bitr3i 290	. . 3 ⊢ ((𝑥 ∈ 𝐵 ∧ ∃𝑦 ∈ 𝐶 𝐴 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
6	5	exbii 1774	. 2 ⊢ (∃𝑥(𝑥 ∈ 𝐵 ∧ ∃𝑦 ∈ 𝐶 𝐴 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
7		df-rex 2918	. 2 ⊢ (∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝐴 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ ∃𝑦 ∈ 𝐶 𝐴 = ⟨𝑥, 𝑦⟩))
8		elxp 5131	. 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
9	6, 7, 8	3bitr4ri 293	1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝐴 = ⟨𝑥, 𝑦⟩)

Syntax hints:   ↔ wb 196   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∃wrex 2913  ⟨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-rex 2918  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: (None)