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Theorem iswun 10745
Description: Properties of a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.)
Assertion
Ref Expression
iswun (𝑈𝑉 → (𝑈 ∈ WUni ↔ (Tr 𝑈𝑈 ≠ ∅ ∧ ∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈))))
Distinct variable group:   𝑥,𝑦,𝑈
Allowed substitution hints:   𝑉(𝑥,𝑦)

Proof of Theorem iswun
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 treq 5266 . . 3 (𝑢 = 𝑈 → (Tr 𝑢 ↔ Tr 𝑈))
2 neeq1 3002 . . 3 (𝑢 = 𝑈 → (𝑢 ≠ ∅ ↔ 𝑈 ≠ ∅))
3 eleq2 2829 . . . . 5 (𝑢 = 𝑈 → ( 𝑥𝑢 𝑥𝑈))
4 eleq2 2829 . . . . 5 (𝑢 = 𝑈 → (𝒫 𝑥𝑢 ↔ 𝒫 𝑥𝑈))
5 eleq2 2829 . . . . . 6 (𝑢 = 𝑈 → ({𝑥, 𝑦} ∈ 𝑢 ↔ {𝑥, 𝑦} ∈ 𝑈))
65raleqbi1dv 3337 . . . . 5 (𝑢 = 𝑈 → (∀𝑦𝑢 {𝑥, 𝑦} ∈ 𝑢 ↔ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈))
73, 4, 63anbi123d 1437 . . . 4 (𝑢 = 𝑈 → (( 𝑥𝑢 ∧ 𝒫 𝑥𝑢 ∧ ∀𝑦𝑢 {𝑥, 𝑦} ∈ 𝑢) ↔ ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈)))
87raleqbi1dv 3337 . . 3 (𝑢 = 𝑈 → (∀𝑥𝑢 ( 𝑥𝑢 ∧ 𝒫 𝑥𝑢 ∧ ∀𝑦𝑢 {𝑥, 𝑦} ∈ 𝑢) ↔ ∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈)))
91, 2, 83anbi123d 1437 . 2 (𝑢 = 𝑈 → ((Tr 𝑢𝑢 ≠ ∅ ∧ ∀𝑥𝑢 ( 𝑥𝑢 ∧ 𝒫 𝑥𝑢 ∧ ∀𝑦𝑢 {𝑥, 𝑦} ∈ 𝑢)) ↔ (Tr 𝑈𝑈 ≠ ∅ ∧ ∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈))))
10 df-wun 10743 . 2 WUni = {𝑢 ∣ (Tr 𝑢𝑢 ≠ ∅ ∧ ∀𝑥𝑢 ( 𝑥𝑢 ∧ 𝒫 𝑥𝑢 ∧ ∀𝑦𝑢 {𝑥, 𝑦} ∈ 𝑢))}
119, 10elab2g 3679 1 (𝑈𝑉 → (𝑈 ∈ WUni ↔ (Tr 𝑈𝑈 ≠ ∅ ∧ ∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  w3a 1086   = wceq 1539  wcel 2107  wne 2939  wral 3060  c0 4332  𝒫 cpw 4599  {cpr 4627   cuni 4906  Tr wtr 5258  WUnicwun 10741
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-v 3481  df-ss 3967  df-uni 4907  df-tr 5259  df-wun 10743
This theorem is referenced by:  wuntr  10746  wununi  10747  wunpw  10748  wunpr  10750  wun0  10759  intwun  10776  r1limwun  10777  wunex2  10779  tskwun  10825  gruwun  10854  pwinfi2  43580
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