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Theorem wuntp 10705
Description: A weak universe is closed under unordered triple. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
wununi.1 (𝜑𝑈 ∈ WUni)
wununi.2 (𝜑𝐴𝑈)
wunpr.3 (𝜑𝐵𝑈)
wuntp.3 (𝜑𝐶𝑈)
Assertion
Ref Expression
wuntp (𝜑 → {𝐴, 𝐵, 𝐶} ∈ 𝑈)

Proof of Theorem wuntp
StepHypRef Expression
1 tpass 4756 . 2 {𝐴, 𝐵, 𝐶} = ({𝐴} ∪ {𝐵, 𝐶})
2 wununi.1 . . 3 (𝜑𝑈 ∈ WUni)
3 dfsn2 4641 . . . 4 {𝐴} = {𝐴, 𝐴}
4 wununi.2 . . . . 5 (𝜑𝐴𝑈)
52, 4, 4wunpr 10703 . . . 4 (𝜑 → {𝐴, 𝐴} ∈ 𝑈)
63, 5eqeltrid 2837 . . 3 (𝜑 → {𝐴} ∈ 𝑈)
7 wunpr.3 . . . 4 (𝜑𝐵𝑈)
8 wuntp.3 . . . 4 (𝜑𝐶𝑈)
92, 7, 8wunpr 10703 . . 3 (𝜑 → {𝐵, 𝐶} ∈ 𝑈)
102, 6, 9wunun 10704 . 2 (𝜑 → ({𝐴} ∪ {𝐵, 𝐶}) ∈ 𝑈)
111, 10eqeltrid 2837 1 (𝜑 → {𝐴, 𝐵, 𝐶} ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106  cun 3946  {csn 4628  {cpr 4630  {ctp 4632  WUnicwun 10694
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-3or 1088  df-3an 1089  df-tru 1544  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-un 3953  df-in 3955  df-ss 3965  df-sn 4629  df-pr 4631  df-tp 4633  df-uni 4909  df-tr 5266  df-wun 10696
This theorem is referenced by:  catcfuccl  18068  catcfucclOLD  18069  catcxpccl  18158  catcxpcclOLD  18159
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