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Theorem wuntp 10204
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 4640 . 2 {𝐴, 𝐵, 𝐶} = ({𝐴} ∪ {𝐵, 𝐶})
2 wununi.1 . . 3 (𝜑𝑈 ∈ WUni)
3 dfsn2 4526 . . . 4 {𝐴} = {𝐴, 𝐴}
4 wununi.2 . . . . 5 (𝜑𝐴𝑈)
52, 4, 4wunpr 10202 . . . 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 10202 . . 3 (𝜑 → {𝐵, 𝐶} ∈ 𝑈)
102, 6, 9wunun 10203 . 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 2113  cun 3839  {csn 4513  {cpr 4515  {ctp 4517  WUnicwun 10193
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 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-ex 1787  df-sb 2074  df-clab 2717  df-cleq 2730  df-clel 2811  df-ne 2935  df-ral 3058  df-v 3399  df-un 3846  df-in 3848  df-ss 3858  df-sn 4514  df-pr 4516  df-tp 4518  df-uni 4794  df-tr 5134  df-wun 10195
This theorem is referenced by:  catcfuccl  17478  catcxpccl  17566
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