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Theorem wunop 10719
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 4871 . . 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 10713 . . 3 (𝜑 → {𝐴} ∈ 𝑈)
75, 1, 2wunpr 10706 . . 3 (𝜑 → {𝐴, 𝐵} ∈ 𝑈)
85, 6, 7wunpr 10706 . 2 (𝜑 → {{𝐴}, {𝐴, 𝐵}} ∈ 𝑈)
94, 8eqeltrd 2833 1 (𝜑 → ⟨𝐴, 𝐵⟩ ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  {csn 4628  {cpr 4630  cop 4634  WUnicwun 10697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-tr 5266  df-wun 10699
This theorem is referenced by:  wunot  10720  1strwunbndx  17167  wunress  17199  wunressOLD  17200  catcoppccl  18071  catcoppcclOLD  18072  catcfuccl  18073  catcfucclOLD  18074  catcxpccl  18163  catcxpcclOLD  18164
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