Theorem wunop 10225

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 4758	. . 3 ⊢ ((𝐴 ∈ 𝑈 ∧ 𝐵 ∈ 𝑈) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
4	1, 2, 3	syl2anc 587	. 2 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
5		wun0.1	. . 3 ⊢ (𝜑 → 𝑈 ∈ WUni)
6	5, 1	wunsn 10219	. . 3 ⊢ (𝜑 → {𝐴} ∈ 𝑈)
7	5, 1, 2	wunpr 10212	. . 3 ⊢ (𝜑 → {𝐴, 𝐵} ∈ 𝑈)
8	5, 6, 7	wunpr 10212	. 2 ⊢ (𝜑 → {{𝐴}, {𝐴, 𝐵}} ∈ 𝑈)
9	4, 8	eqeltrd 2834	1 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ 𝑈)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  {csn 4517  {cpr 4519  ⟨cop 4523  WUnicwun 10203

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2711

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-ne 2936  df-ral 3059  df-v 3401  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-if 4416  df-sn 4518  df-pr 4520  df-op 4524  df-uni 4798  df-tr 5138  df-wun 10205

This theorem is referenced by:  wunot  10226  wunress  16670  1strwunbndx  16706  catcoppccl  17487  catcfuccl  17488  catcxpccl  17576