Theorem xpmlem 5090

Description: The cross product of inhabited classes is inhabited. (Contributed by Jim Kingdon, 11-Dec-2018.)

Assertion

Ref	Expression
xpmlem	⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 ∈ 𝐵) ↔ ∃𝑧 𝑧 ∈ (𝐴 × 𝐵))

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

Proof of Theorem xpmlem

Step	Hyp	Ref	Expression
1		eeanv 1951	. . 3 ⊢ (∃𝑥∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (∃𝑥 𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 ∈ 𝐵))
2		vex 2766	. . . . . 6 ⊢ 𝑥 ∈ V
3		vex 2766	. . . . . 6 ⊢ 𝑦 ∈ V
4	2, 3	opex 4262	. . . . 5 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
5		eleq1 2259	. . . . . 6 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ (𝐴 × 𝐵) ↔ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)))
6		opelxp 4693	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
7	5, 6	bitrdi 196	. . . . 5 ⊢ (𝑧 = ⟨𝑥, 𝑦⟩ → (𝑧 ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
8	4, 7	spcev 2859	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ∃𝑧 𝑧 ∈ (𝐴 × 𝐵))
9	8	exlimivv 1911	. . 3 ⊢ (∃𝑥∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ∃𝑧 𝑧 ∈ (𝐴 × 𝐵))
10	1, 9	sylbir 135	. 2 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 ∈ 𝐵) → ∃𝑧 𝑧 ∈ (𝐴 × 𝐵))
11		elxp 4680	. . . . 5 ⊢ (𝑧 ∈ (𝐴 × 𝐵) ↔ ∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
12		simpr 110	. . . . . 6 ⊢ ((𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
13	12	2eximi 1615	. . . . 5 ⊢ (∃𝑥∃𝑦(𝑧 = ⟨𝑥, 𝑦⟩ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ∃𝑥∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
14	11, 13	sylbi 121	. . . 4 ⊢ (𝑧 ∈ (𝐴 × 𝐵) → ∃𝑥∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
15	14	exlimiv 1612	. . 3 ⊢ (∃𝑧 𝑧 ∈ (𝐴 × 𝐵) → ∃𝑥∃𝑦(𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
16	15, 1	sylib 122	. 2 ⊢ (∃𝑧 𝑧 ∈ (𝐴 × 𝐵) → (∃𝑥 𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 ∈ 𝐵))
17	10, 16	impbii 126	1 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 ∈ 𝐵) ↔ ∃𝑧 𝑧 ∈ (𝐴 × 𝐵))

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1364  ∃wex 1506   ∈ wcel 2167  ⟨cop 3625   × cxp 4661

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-opab 4095  df-xp 4669

This theorem is referenced by:  xpm  5091