Theorem wunop 10681

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 4830	. . 3 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
4	1, 2, 3	syl2anc 593	. 2 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
5		wun0.1	. . 3 ⊢ (𝜑 → 𝑈 ∈ WUni)
6	5, 1	wunsn 10675	. . 3 ⊢ (𝜑 → {𝐴} ∈ 𝑈)
7	5, 1, 2	wunpr 10668	. . 3 ⊢ (𝜑 → {𝐴, 𝐵} ∈ 𝑈)
8	5, 6, 7	wunpr 10668	. 2 ⊢ (𝜑 → {{𝐴}, {𝐴, 𝐵}} ∈ 𝑈)
9	4, 8	eqeltrd 2863	1 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ 𝑈)

Syntax hints:   → wi 4   = wceq 1561   ∈ wcel 2143  {csn 4583  {cpr 4585  ⟨cop 4589  WUnicwun 10659

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-sb 2092  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3078  df-rex 3088  df-v 3457  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-tr 5209  df-wun 10661

This theorem is referenced by:  wunot  10682  1strwunbndx  17262  wunress  17286  catcoppccl  18151  catcfuccl  18152  catcxpccl  18240