Theorem xpm 5087

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 5086	. 2 ⊢ ((∃𝑎 𝑎 ∈ 𝐴 ∧ ∃𝑏 𝑏 ∈ 𝐵) ↔ ∃𝑤 𝑤 ∈ (𝐴 × 𝐵))
2		eleq1 2256	. . . 4 ⊢ (𝑎 = 𝑥 → (𝑎 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
3	2	cbvexv 1930	. . 3 ⊢ (∃𝑎 𝑎 ∈ 𝐴 ↔ ∃𝑥 𝑥 ∈ 𝐴)
4		eleq1 2256	. . . 4 ⊢ (𝑏 = 𝑦 → (𝑏 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵))
5	4	cbvexv 1930	. . 3 ⊢ (∃𝑏 𝑏 ∈ 𝐵 ↔ ∃𝑦 𝑦 ∈ 𝐵)
6	3, 5	anbi12i 460	. 2 ⊢ ((∃𝑎 𝑎 ∈ 𝐴 ∧ ∃𝑏 𝑏 ∈ 𝐵) ↔ (∃𝑥 𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 ∈ 𝐵))
7		eleq1 2256	. . 3 ⊢ (𝑤 = 𝑧 → (𝑤 ∈ (𝐴 × 𝐵) ↔ 𝑧 ∈ (𝐴 × 𝐵)))
8	7	cbvexv 1930	. 2 ⊢ (∃𝑤 𝑤 ∈ (𝐴 × 𝐵) ↔ ∃𝑧 𝑧 ∈ (𝐴 × 𝐵))
9	1, 6, 8	3bitr3i 210	1 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 ∈ 𝐵) ↔ ∃𝑧 𝑧 ∈ (𝐴 × 𝐵))

Syntax hints:   ∧ wa 104   ↔ wb 105  ∃wex 1503   ∈ wcel 2164   × cxp 4657

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-opab 4091  df-xp 4665

This theorem is referenced by:  ssxpbm  5101  xp11m  5104  xpexr2m  5107  unixpm  5201