Theorem xpm 5025

Description: The cross product of inhabited classes is inhabited. (Contributed by Jim Kingdon, 13-Dec-2018.)

Assertion

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

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

Allowed substitution hints:   𝐴(𝑦)   𝐵(𝑥)

Proof of Theorem xpm

Dummy variables 𝑎 𝑏 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpmlem 5024	. 2 ⊢ ((∃𝑎 𝑎 ∈ 𝐴 ∧ ∃𝑏 𝑏 ∈ 𝐵) ↔ ∃𝑤 𝑤 ∈ (𝐴 × 𝐵))
2		eleq1 2229	. . . 4 ⊢ (𝑎 = 𝑥 → (𝑎 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
3	2	cbvexv 1906	. . 3 ⊢ (∃𝑎 𝑎 ∈ 𝐴 ↔ ∃𝑥 𝑥 ∈ 𝐴)
4		eleq1 2229	. . . 4 ⊢ (𝑏 = 𝑦 → (𝑏 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵))
5	4	cbvexv 1906	. . 3 ⊢ (∃𝑏 𝑏 ∈ 𝐵 ↔ ∃𝑦 𝑦 ∈ 𝐵)
6	3, 5	anbi12i 456	. 2 ⊢ ((∃𝑎 𝑎 ∈ 𝐴 ∧ ∃𝑏 𝑏 ∈ 𝐵) ↔ (∃𝑥 𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 ∈ 𝐵))
7		eleq1 2229	. . 3 ⊢ (𝑤 = 𝑧 → (𝑤 ∈ (𝐴 × 𝐵) ↔ 𝑧 ∈ (𝐴 × 𝐵)))
8	7	cbvexv 1906	. 2 ⊢ (∃𝑤 𝑤 ∈ (𝐴 × 𝐵) ↔ ∃𝑧 𝑧 ∈ (𝐴 × 𝐵))
9	1, 6, 8	3bitr3i 209	1 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 ∈ 𝐵) ↔ ∃𝑧 𝑧 ∈ (𝐴 × 𝐵))

Syntax hints:   ∧ wa 103   ↔ wb 104  ∃wex 1480   ∈ wcel 2136   × 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-ral 2449  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:  ssxpbm  5039  xp11m  5042  xpexr2m  5045  unixpm  5139