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Theorem wunop 10478
Description: A weak universe is closed under ordered pairs. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
wun0.1 (𝜑𝑈 ∈ WUni)
wunop.2 (𝜑𝐴𝑈)
wunop.3 (𝜑𝐵𝑈)
Assertion
Ref Expression
wunop (𝜑 → ⟨𝐴, 𝐵⟩ ∈ 𝑈)

Proof of Theorem wunop
StepHypRef Expression
1 wunop.2 . . 3 (𝜑𝐴𝑈)
2 wunop.3 . . 3 (𝜑𝐵𝑈)
3 dfopg 4802 . . 3 ((𝐴𝑈𝐵𝑈) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
41, 2, 3syl2anc 584 . 2 (𝜑 → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
5 wun0.1 . . 3 (𝜑𝑈 ∈ WUni)
65, 1wunsn 10472 . . 3 (𝜑 → {𝐴} ∈ 𝑈)
75, 1, 2wunpr 10465 . . 3 (𝜑 → {𝐴, 𝐵} ∈ 𝑈)
85, 6, 7wunpr 10465 . 2 (𝜑 → {{𝐴}, {𝐴, 𝐵}} ∈ 𝑈)
94, 8eqeltrd 2839 1 (𝜑 → ⟨𝐴, 𝐵⟩ ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2106  {csn 4561  {cpr 4563  cop 4567  WUnicwun 10456
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-tr 5192  df-wun 10458
This theorem is referenced by:  wunot  10479  1strwunbndx  16931  wunress  16960  wunressOLD  16961  catcoppccl  17832  catcoppcclOLD  17833  catcfuccl  17834  catcfucclOLD  17835  catcxpccl  17924  catcxpcclOLD  17925
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