Theorem xpexd 4843

Description: The Cartesian product of two sets is a set. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
xpexd.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
xpexd.2	⊢ (𝜑 → 𝐵 ∈ 𝑊)

Assertion

Ref	Expression
xpexd	⊢ (𝜑 → (𝐴 × 𝐵) ∈ V)

Proof of Theorem xpexd

Step	Hyp	Ref	Expression
1		xpexd.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		xpexd.2	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
3		xpexg 4842	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 × 𝐵) ∈ V)
4	1, 2, 3	syl2anc 411	1 ⊢ (𝜑 → (𝐴 × 𝐵) ∈ V)

Syntax hints:   → wi 4   ∈ wcel 2201  Vcvv 2801   × cxp 4725

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2203  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266  ax-pr 4301  ax-un 4532

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-rex 2515  df-v 2803  df-un 3203  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-opab 4152  df-xp 4733

This theorem is referenced by:  opabex2  6362