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Theorem wuntp 10121
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 4680 . 2 {𝐴, 𝐵, 𝐶} = ({𝐴} ∪ {𝐵, 𝐶})
2 wununi.1 . . 3 (𝜑𝑈 ∈ WUni)
3 dfsn2 4570 . . . 4 {𝐴} = {𝐴, 𝐴}
4 wununi.2 . . . . 5 (𝜑𝐴𝑈)
52, 4, 4wunpr 10119 . . . 4 (𝜑 → {𝐴, 𝐴} ∈ 𝑈)
63, 5eqeltrid 2914 . . 3 (𝜑 → {𝐴} ∈ 𝑈)
7 wunpr.3 . . . 4 (𝜑𝐵𝑈)
8 wuntp.3 . . . 4 (𝜑𝐶𝑈)
92, 7, 8wunpr 10119 . . 3 (𝜑 → {𝐵, 𝐶} ∈ 𝑈)
102, 6, 9wunun 10120 . 2 (𝜑 → ({𝐴} ∪ {𝐵, 𝐶}) ∈ 𝑈)
111, 10eqeltrid 2914 1 (𝜑 → {𝐴, 𝐵, 𝐶} ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2105  cun 3931  {csn 4557  {cpr 4559  {ctp 4561  WUnicwun 10110
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-v 3494  df-un 3938  df-in 3940  df-ss 3949  df-sn 4558  df-pr 4560  df-tp 4562  df-uni 4831  df-tr 5164  df-wun 10112
This theorem is referenced by:  catcfuccl  17357  catcxpccl  17445
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