Theorem wunop 10736

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 4847	. . 3 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
4	1, 2, 3	syl2anc 584	. 2 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
5		wun0.1	. . 3 ⊢ (𝜑 → 𝑈 ∈ WUni)
6	5, 1	wunsn 10730	. . 3 ⊢ (𝜑 → {𝐴} ∈ 𝑈)
7	5, 1, 2	wunpr 10723	. . 3 ⊢ (𝜑 → {𝐴, 𝐵} ∈ 𝑈)
8	5, 6, 7	wunpr 10723	. 2 ⊢ (𝜑 → {{𝐴}, {𝐴, 𝐵}} ∈ 𝑈)
9	4, 8	eqeltrd 2834	1 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ 𝑈)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  {csn 4601  {cpr 4603  ⟨cop 4607  WUnicwun 10714

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-ral 3052  df-rex 3061  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-tr 5230  df-wun 10716

This theorem is referenced by:  wunot  10737  1strwunbndx  17246  wunress  17270  catcoppccl  18130  catcfuccl  18131  catcxpccl  18219