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Theorem wuntp 9477
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 4257 . 2 {𝐴, 𝐵, 𝐶} = ({𝐴} ∪ {𝐵, 𝐶})
2 wununi.1 . . 3 (𝜑𝑈 ∈ WUni)
3 dfsn2 4161 . . . 4 {𝐴} = {𝐴, 𝐴}
4 wununi.2 . . . . 5 (𝜑𝐴𝑈)
52, 4, 4wunpr 9475 . . . 4 (𝜑 → {𝐴, 𝐴} ∈ 𝑈)
63, 5syl5eqel 2702 . . 3 (𝜑 → {𝐴} ∈ 𝑈)
7 wunpr.3 . . . 4 (𝜑𝐵𝑈)
8 wuntp.3 . . . 4 (𝜑𝐶𝑈)
92, 7, 8wunpr 9475 . . 3 (𝜑 → {𝐵, 𝐶} ∈ 𝑈)
102, 6, 9wunun 9476 . 2 (𝜑 → ({𝐴} ∪ {𝐵, 𝐶}) ∈ 𝑈)
111, 10syl5eqel 2702 1 (𝜑 → {𝐴, 𝐵, 𝐶} ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1987  cun 3553  {csn 4148  {cpr 4150  {ctp 4152  WUnicwun 9466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-v 3188  df-un 3560  df-in 3562  df-ss 3569  df-sn 4149  df-pr 4151  df-tp 4153  df-uni 4403  df-tr 4713  df-wun 9468
This theorem is referenced by:  catcfuccl  16680  catcxpccl  16768
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