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Theorem wunop 10138
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 4795 . . 3 ((𝐴𝑈𝐵𝑈) → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
41, 2, 3syl2anc 586 . 2 (𝜑 → ⟨𝐴, 𝐵⟩ = {{𝐴}, {𝐴, 𝐵}})
5 wun0.1 . . 3 (𝜑𝑈 ∈ WUni)
65, 1wunsn 10132 . . 3 (𝜑 → {𝐴} ∈ 𝑈)
75, 1, 2wunpr 10125 . . 3 (𝜑 → {𝐴, 𝐵} ∈ 𝑈)
85, 6, 7wunpr 10125 . 2 (𝜑 → {{𝐴}, {𝐴, 𝐵}} ∈ 𝑈)
94, 8eqeltrd 2913 1 (𝜑 → ⟨𝐴, 𝐵⟩ ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2110  {csn 4561  {cpr 4563  cop 4567  WUnicwun 10116
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-v 3497  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-tr 5166  df-wun 10118
This theorem is referenced by:  wunot  10139  wunress  16558  1strwunbndx  16594  catcoppccl  17362  catcfuccl  17363  catcxpccl  17451
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