MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  wunpr Structured version   Visualization version   GIF version

Theorem wunpr 9784
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 9779 . . . . 5 (𝑈 ∈ WUni → (𝑈 ∈ WUni ↔ (Tr 𝑈𝑈 ≠ ∅ ∧ ∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈))))
54ibi 258 . . . 4 (𝑈 ∈ WUni → (Tr 𝑈𝑈 ≠ ∅ ∧ ∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈)))
65simp3d 1174 . . 3 (𝑈 ∈ WUni → ∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈))
7 simp3 1168 . . . 4 (( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈) → ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈)
87ralimi 3099 . . 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 4423 . . . 4 (𝑥 = 𝐴 → {𝑥, 𝑦} = {𝐴, 𝑦})
1110eleq1d 2829 . . 3 (𝑥 = 𝐴 → ({𝑥, 𝑦} ∈ 𝑈 ↔ {𝐴, 𝑦} ∈ 𝑈))
12 preq2 4424 . . . 4 (𝑦 = 𝐵 → {𝐴, 𝑦} = {𝐴, 𝐵})
1312eleq1d 2829 . . 3 (𝑦 = 𝐵 → ({𝐴, 𝑦} ∈ 𝑈 ↔ {𝐴, 𝐵} ∈ 𝑈))
1411, 13rspc2va 3475 . 2 (((𝐴𝑈𝐵𝑈) ∧ ∀𝑥𝑈𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈) → {𝐴, 𝐵} ∈ 𝑈)
151, 2, 9, 14syl21anc 866 1 (𝜑 → {𝐴, 𝐵} ∈ 𝑈)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1107   = wceq 1652  wcel 2155  wne 2937  wral 3055  c0 4079  𝒫 cpw 4315  {cpr 4336   cuni 4594  Tr wtr 4911  WUnicwun 9775
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-v 3352  df-un 3737  df-in 3739  df-ss 3746  df-sn 4335  df-pr 4337  df-uni 4595  df-tr 4912  df-wun 9777
This theorem is referenced by:  wunun  9785  wuntp  9786  wunsn  9791  wunop  9797  intwun  9810  wuncval2  9822
  Copyright terms: Public domain W3C validator