Theorem elxp2 4767

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 2526	. . . 4 ⊢ (∃𝑦 ∈ 𝐶 (𝑥 ∈ 𝐵 ∧ 𝐴 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑦(𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐵 ∧ 𝐴 = ⟨𝑥, 𝑦⟩)))
2		r19.42v 2700	. . . 4 ⊢ (∃𝑦 ∈ 𝐶 (𝑥 ∈ 𝐵 ∧ 𝐴 = ⟨𝑥, 𝑦⟩) ↔ (𝑥 ∈ 𝐵 ∧ ∃𝑦 ∈ 𝐶 𝐴 = ⟨𝑥, 𝑦⟩))
3		an13 565	. . . . 5 ⊢ ((𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐵 ∧ 𝐴 = ⟨𝑥, 𝑦⟩)) ↔ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
4	3	exbii 1654	. . . 4 ⊢ (∃𝑦(𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐵 ∧ 𝐴 = ⟨𝑥, 𝑦⟩)) ↔ ∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
5	1, 2, 4	3bitr3i 210	. . 3 ⊢ ((𝑥 ∈ 𝐵 ∧ ∃𝑦 ∈ 𝐶 𝐴 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
6	5	exbii 1654	. 2 ⊢ (∃𝑥(𝑥 ∈ 𝐵 ∧ ∃𝑦 ∈ 𝐶 𝐴 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
7		df-rex 2526	. 2 ⊢ (∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝐴 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ ∃𝑦 ∈ 𝐶 𝐴 = ⟨𝑥, 𝑦⟩))
8		elxp 4766	. 2 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
9	6, 7, 8	3bitr4ri 213	1 ⊢ (𝐴 ∈ (𝐵 × 𝐶) ↔ ∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝐴 = ⟨𝑥, 𝑦⟩)

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1398  ∃wex 1541   ∈ wcel 2203  ∃wrex 2521  ⟨cop 3692   × cxp 4747

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-opab 4172  df-xp 4755

This theorem is referenced by:  opelxp  4779  xpiundi  4808  xpiundir  4809  ssrel2  4840  f1o2ndf1  6424  xpdom2  7082  elreal  8143