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Theorem wunop 10225
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 4758 . . 3 ((𝐴𝑈𝐵𝑈) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
41, 2, 3syl2anc 587 . 2 (𝜑 → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
5 wun0.1 . . 3 (𝜑𝑈 ∈ WUni)
65, 1wunsn 10219 . . 3 (𝜑 → {𝐴} ∈ 𝑈)
75, 1, 2wunpr 10212 . . 3 (𝜑 → {𝐴, 𝐵} ∈ 𝑈)
85, 6, 7wunpr 10212 . 2 (𝜑 → {{𝐴}, {𝐴, 𝐵}} ∈ 𝑈)
94, 8eqeltrd 2834 1 (𝜑 → ⟨𝐴, 𝐵⟩ ∈ 𝑈)
Colors of variables: wff setvar class
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
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