Theorem wunxp 10676

Description: A weak universe is closed under cartesian products. (Contributed by Mario Carneiro, 2-Jan-2017.)

Hypotheses

Ref	Expression
wun0.1	⊢ (𝜑 → 𝑈 ∈ WUni)
wunop.2	⊢ (𝜑 → 𝐴 ∈ 𝑈)
wunop.3	⊢ (𝜑 → 𝐵 ∈ 𝑈)

Assertion

Ref	Expression
wunxp	⊢ (𝜑 → (𝐴 × 𝐵) ∈ 𝑈)

Proof of Theorem wunxp

Step	Hyp	Ref	Expression
1		wun0.1	. 2 ⊢ (𝜑 → 𝑈 ∈ WUni)
2		wunop.2	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
3		wunop.3	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑈)
4	1, 2, 3	wunun 10662	. . . 4 ⊢ (𝜑 → (𝐴 ∪ 𝐵) ∈ 𝑈)
5	1, 4	wunpw 10659	. . 3 ⊢ (𝜑 → 𝒫 (𝐴 ∪ 𝐵) ∈ 𝑈)
6	1, 5	wunpw 10659	. 2 ⊢ (𝜑 → 𝒫 𝒫 (𝐴 ∪ 𝐵) ∈ 𝑈)
7		xpsspw 5778	. . 3 ⊢ (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝐵)
8	7	a1i 11	. 2 ⊢ (𝜑 → (𝐴 × 𝐵) ⊆ 𝒫 𝒫 (𝐴 ∪ 𝐵))
9	1, 6, 8	wunss 10664	1 ⊢ (𝜑 → (𝐴 × 𝐵) ∈ 𝑈)

Syntax hints:   → wi 4   ∈ wcel 2141   ∪ cun 3900   ⊆ wss 3902  𝒫 cpw 4552   × cxp 5641  WUnicwun 10652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5243  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-opab 5160  df-tr 5205  df-xp 5649  df-rel 5650  df-wun 10654

This theorem is referenced by:  wunpm  10677  wuncnv  10682  wunco  10685  wuntpos  10686  tskxp  10739  wuncn  11122  wunfunc  17925  wunnat  17983  catcoppccl  18141  catcfuccl  18142  catcxpccl  18230