Theorem wunop 10409

Description: A weak universe is closed under ordered pairs. (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
wunop	⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ 𝑈)

Proof of Theorem wunop

Step	Hyp	Ref	Expression
1		wunop.2	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
2		wunop.3	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑈)
3		dfopg 4799	. . 3 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
4	1, 2, 3	syl2anc 583	. 2 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
5		wun0.1	. . 3 ⊢ (𝜑 → 𝑈 ∈ WUni)
6	5, 1	wunsn 10403	. . 3 ⊢ (𝜑 → {𝐴} ∈ 𝑈)
7	5, 1, 2	wunpr 10396	. . 3 ⊢ (𝜑 → {𝐴, 𝐵} ∈ 𝑈)
8	5, 6, 7	wunpr 10396	. 2 ⊢ (𝜑 → {{𝐴}, {𝐴, 𝐵}} ∈ 𝑈)
9	4, 8	eqeltrd 2839	1 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ 𝑈)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  {csn 4558  {cpr 4560  ⟨cop 4564  WUnicwun 10387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-ral 3068  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-tr 5188  df-wun 10389

This theorem is referenced by:  wunot  10410  1strwunbndx  16857  wunress  16886  wunressOLD  16887  catcoppccl  17748  catcoppcclOLD  17749  catcfuccl  17750  catcfucclOLD  17751  catcxpccl  17840  catcxpcclOLD  17841