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Theorem wunpr 9737
Description: A weak universe is closed under pairing. (Contributed by Mario Carneiro, 2-Jan-2017.)
Hypotheses
Ref Expression
wununi.1 (𝜑𝑈 ∈ WUni)
wununi.2 (𝜑𝐴𝑈)
wunpr.3 (𝜑𝐵𝑈)
Assertion
Ref Expression
wunpr (𝜑 → {𝐴, 𝐵} ∈ 𝑈)

Proof of Theorem wunpr
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 wununi.2 . 2 (𝜑𝐴𝑈)
2 wunpr.3 . 2 (𝜑𝐵𝑈)
3 wununi.1 . . 3 (𝜑𝑈 ∈ WUni)
4 iswun 9732 . . . . 5 (𝑈 ∈ WUni → (𝑈 ∈ WUni ↔ (Tr 𝑈𝑈 ≠ ∅ ∧ ∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈))))
54ibi 256 . . . 4 (𝑈 ∈ WUni → (Tr 𝑈𝑈 ≠ ∅ ∧ ∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈)))
65simp3d 1138 . . 3 (𝑈 ∈ WUni → ∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈))
7 simp3 1132 . . . 4 (( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈) → ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈)
87ralimi 3101 . . 3 (∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈) → ∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈)
93, 6, 83syl 18 . 2 (𝜑 → ∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈)
10 preq1 4405 . . . 4 (𝑥 = 𝐴 → {𝑥, 𝑦} = {𝐴, 𝑦})
1110eleq1d 2835 . . 3 (𝑥 = 𝐴 → ({𝑥, 𝑦} ∈ 𝑈 ↔ {𝐴, 𝑦} ∈ 𝑈))
12 preq2 4406 . . . 4 (𝑦 = 𝐵 → {𝐴, 𝑦} = {𝐴, 𝐵})
1312eleq1d 2835 . . 3 (𝑦 = 𝐵 → ({𝐴, 𝑦} ∈ 𝑈 ↔ {𝐴, 𝐵} ∈ 𝑈))
1411, 13rspc2va 3473 . 2 (((𝐴𝑈𝐵𝑈) ∧ ∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈) → {𝐴, 𝐵} ∈ 𝑈)
151, 2, 9, 14syl21anc 1475 1 (𝜑 → {𝐴, 𝐵} ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1071   = wceq 1631  wcel 2145  wne 2943  wral 3061  c0 4063  𝒫 cpw 4298  {cpr 4319   cuni 4575  Tr wtr 4887  WUnicwun 9728
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-v 3353  df-un 3728  df-in 3730  df-ss 3737  df-sn 4318  df-pr 4320  df-uni 4576  df-tr 4888  df-wun 9730
This theorem is referenced by:  wunun  9738  wuntp  9739  wunsn  9744  wunop  9750  intwun  9763  wuncval2  9775
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