Theorem elxp2 4622

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 2450	. . . 4 ⊢ (∃𝑦 ∈ 𝐶 (𝑥 ∈ 𝐵 ∧ 𝐴 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑦(𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐵 ∧ 𝐴 = ⟨𝑥, 𝑦⟩)))
2		r19.42v 2623	. . . 4 ⊢ (∃𝑦 ∈ 𝐶 (𝑥 ∈ 𝐵 ∧ 𝐴 = ⟨𝑥, 𝑦⟩) ↔ (𝑥 ∈ 𝐵 ∧ ∃𝑦 ∈ 𝐶 𝐴 = ⟨𝑥, 𝑦⟩))
3		an13 553	. . . . 5 ⊢ ((𝑦 ∈ 𝐶 ∧ (𝑥 ∈ 𝐵 ∧ 𝐴 = ⟨𝑥, 𝑦⟩)) ↔ (𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
4	3	exbii 1593	. . . 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 1593	. 2 ⊢ (∃𝑥(𝑥 ∈ 𝐵 ∧ ∃𝑦 ∈ 𝐶 𝐴 = ⟨𝑥, 𝑦⟩) ↔ ∃𝑥∃𝑦(𝐴 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐶)))
7		df-rex 2450	. 2 ⊢ (∃𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 𝐴 = ⟨𝑥, 𝑦⟩ ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ ∃𝑦 ∈ 𝐶 𝐴 = ⟨𝑥, 𝑦⟩))
8		elxp 4621	. 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 1343  ∃wex 1480   ∈ wcel 2136  ∃wrex 2445  ⟨cop 3579   × cxp 4602

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-opab 4044  df-xp 4610

This theorem is referenced by:  opelxp  4634  xpiundi  4662  xpiundir  4663  ssrel2  4694  f1o2ndf1  6196  xpdom2  6797  elreal  7769