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Theorem wuntp 9412
 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 4231 . 2 {𝐴, 𝐵, 𝐶} = ({𝐴} ∪ {𝐵, 𝐶})
2 wununi.1 . . 3 (𝜑𝑈 ∈ WUni)
3 dfsn2 4138 . . . 4 {𝐴} = {𝐴, 𝐴}
4 wununi.2 . . . . 5 (𝜑𝐴𝑈)
52, 4, 4wunpr 9410 . . . 4 (𝜑 → {𝐴, 𝐴} ∈ 𝑈)
63, 5syl5eqel 2692 . . 3 (𝜑 → {𝐴} ∈ 𝑈)
7 wunpr.3 . . . 4 (𝜑𝐵𝑈)
8 wuntp.3 . . . 4 (𝜑𝐶𝑈)
92, 7, 8wunpr 9410 . . 3 (𝜑 → {𝐵, 𝐶} ∈ 𝑈)
102, 6, 9wunun 9411 . 2 (𝜑 → ({𝐴} ∪ {𝐵, 𝐶}) ∈ 𝑈)
111, 10syl5eqel 2692 1 (𝜑 → {𝐴, 𝐵, 𝐶} ∈ 𝑈)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 1977   ∪ cun 3538  {csn 4125  {cpr 4127  {ctp 4129  WUnicwun 9401 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-v 3175  df-un 3545  df-in 3547  df-ss 3554  df-sn 4126  df-pr 4128  df-tp 4130  df-uni 4373  df-tr 4681  df-wun 9403 This theorem is referenced by:  catcfuccl  16582  catcxpccl  16670
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