Theorem elxp2 4629

Description: Membership in a cross product. (Contributed by NM, 23-Feb-2004.)

Assertion

Ref	Expression
elxp2	⊢ (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 elxp2

Step	Hyp	Ref	Expression
1		df-rex 2454	. . . 4 ⊢ (∃𝑦 ∈ 𝐶 (𝑥 ∈ 𝐵 ∧ 𝐴 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑦(𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐵 ∧ 𝐴 = ⟨𝑥, 𝑦⟩)))
2		r19.42v 2627	. . . 4 ⊢ (∃𝑦 ∈ 𝐶 (𝑥 ∈ 𝐵 ∧ 𝐴 = ⟨𝑥, 𝑦⟩) ↔ (𝑥 ∈ 𝐵 ∧ ∃𝑦 ∈ 𝐶 𝐴 = ⟨𝑥, 𝑦⟩))
3		an13 558	. . . . 5 ⊢ ((𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐵 ∧ 𝐴 = ⟨𝑥, 𝑦⟩)) ↔ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
4	3	exbii 1598	. . . 4 ⊢ (∃𝑦(𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐵 ∧ 𝐴 = ⟨𝑥, 𝑦⟩)) ↔ ∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
5	1, 2, 4	3bitr3i 209	. . 3 ⊢ ((𝑥 ∈ 𝐵 ∧ ∃𝑦 ∈ 𝐶 𝐴 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
6	5	exbii 1598	. 2 ⊢ (∃𝑥(𝑥 ∈ 𝐵 ∧ ∃𝑦 ∈ 𝐶 𝐴 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
7		df-rex 2454	. 2 ⊢ (∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝐴 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ ∃𝑦 ∈ 𝐶 𝐴 = ⟨𝑥, 𝑦⟩))
8		elxp 4628	. 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
9	6, 7, 8	3bitr4ri 212	1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝐴 = ⟨𝑥, 𝑦⟩)

Syntax hints:   ∧ wa 103   ↔ wb 104   = wceq 1348  ∃wex 1485   ∈ wcel 2141  ∃wrex 2449  ⟨cop 3586   × cxp 4609

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-opab 4051  df-xp 4617

This theorem is referenced by:  opelxp  4641  xpiundi  4669  xpiundir  4670  ssrel2  4701  f1o2ndf1  6207  xpdom2  6809  elreal  7790