Theorem xpm 5207

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 5206	. 2 ⊢ ((∃𝑎 𝑎 ∈ 𝐴 ∧ ∃𝑏 𝑏 ∈ 𝐵) ↔ ∃𝑤 𝑤 ∈ (𝐴 × 𝐵))
2		eleq1 2301	. . . 4 ⊢ (𝑎 = 𝑥 → (𝑎 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
3	2	cbvexv 1974	. . 3 ⊢ (∃𝑎 𝑎 ∈ 𝐴 ↔ ∃𝑥 𝑥 ∈ 𝐴)
4		eleq1 2301	. . . 4 ⊢ (𝑏 = 𝑦 → (𝑏 ∈ 𝐵 ↔ 𝑦 ∈ 𝐵))
5	4	cbvexv 1974	. . 3 ⊢ (∃𝑏 𝑏 ∈ 𝐵 ↔ ∃𝑦 𝑦 ∈ 𝐵)
6	3, 5	anbi12i 464	. 2 ⊢ ((∃𝑎 𝑎 ∈ 𝐴 ∧ ∃𝑏 𝑏 ∈ 𝐵) ↔ (∃𝑥 𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 ∈ 𝐵))
7		eleq1 2301	. . 3 ⊢ (𝑤 = 𝑧 → (𝑤 ∈ (𝐴 × 𝐵) ↔ 𝑧 ∈ (𝐴 × 𝐵)))
8	7	cbvexv 1974	. 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 1545   ∈ wcel 2209   × cxp 4770

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-opab 4191  df-xp 4778

This theorem is referenced by:  ssxpbm  5221  xp11m  5224  xpexr2m  5227  unixpm  5321  elmpom  6468