Theorem wunop 10714

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 4864	. . 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 10708	. . 3 ⊢ (𝜑 → {𝐴} ∈ 𝑈)
7	5, 1, 2	wunpr 10701	. . 3 ⊢ (𝜑 → {𝐴, 𝐵} ∈ 𝑈)
8	5, 6, 7	wunpr 10701	. 2 ⊢ (𝜑 → {{𝐴}, {𝐴, 𝐵}} ∈ 𝑈)
9	4, 8	eqeltrd 2825	1 ⊢ (𝜑 → ⟨𝐴, 𝐵⟩ ∈ 𝑈)

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  {csn 4621  {cpr 4623  ⟨cop 4627  WUnicwun 10692

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-ral 3054  df-rex 3063  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-tr 5257  df-wun 10694

This theorem is referenced by:  wunot  10715  1strwunbndx  17164  wunress  17196  wunressOLD  17197  catcoppccl  18071  catcoppcclOLD  18072  catcfuccl  18073  catcfucclOLD  18074  catcxpccl  18163  catcxpcclOLD  18164