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Theorem iswun 9382
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 4680 . . 3 (𝑢 = 𝑈 → (Tr 𝑢 ↔ Tr 𝑈))
2 neeq1 2843 . . 3 (𝑢 = 𝑈 → (𝑢 ≠ ∅ ↔ 𝑈 ≠ ∅))
3 eleq2 2676 . . . . 5 (𝑢 = 𝑈 → ( 𝑥𝑢 𝑥𝑈))
4 eleq2 2676 . . . . 5 (𝑢 = 𝑈 → (𝒫 𝑥𝑢 ↔ 𝒫 𝑥𝑈))
5 eleq2 2676 . . . . . 6 (𝑢 = 𝑈 → ({𝑥, 𝑦} ∈ 𝑢 ↔ {𝑥, 𝑦} ∈ 𝑈))
65raleqbi1dv 3122 . . . . 5 (𝑢 = 𝑈 → (∀𝑦𝑢 {𝑥, 𝑦} ∈ 𝑢 ↔ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈))
73, 4, 63anbi123d 1390 . . . 4 (𝑢 = 𝑈 → (( 𝑥𝑢 ∧ 𝒫 𝑥𝑢 ∧ ∀𝑦𝑢 {𝑥, 𝑦} ∈ 𝑢) ↔ ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈)))
87raleqbi1dv 3122 . . 3 (𝑢 = 𝑈 → (∀𝑥𝑢 ( 𝑥𝑢 ∧ 𝒫 𝑥𝑢 ∧ ∀𝑦𝑢 {𝑥, 𝑦} ∈ 𝑢) ↔ ∀𝑥𝑈 ( 𝑥𝑈 ∧ 𝒫 𝑥𝑈 ∧ ∀𝑦𝑈 {𝑥, 𝑦} ∈ 𝑈)))
91, 2, 83anbi123d 1390 . 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 9380 . 2 WUni = {𝑢 ∣ (Tr 𝑢𝑢 ≠ ∅ ∧ ∀𝑥𝑢 ( 𝑥𝑢 ∧ 𝒫 𝑥𝑢 ∧ ∀𝑦𝑢 {𝑥, 𝑦} ∈ 𝑢))}
119, 10elab2g 3321 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 194  w3a 1030   = wceq 1474  wcel 1976  wne 2779  wral 2895  c0 3873  𝒫 cpw 4107  {cpr 4126   cuni 4366  Tr wtr 4674  WUnicwun 9378
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-v 3174  df-in 3546  df-ss 3553  df-uni 4367  df-tr 4675  df-wun 9380
This theorem is referenced by:  wuntr  9383  wununi  9384  wunpw  9385  wunpr  9387  wun0  9396  intwun  9413  r1limwun  9414  wunex2  9416  tskwun  9462  gruwun  9491  pwinfi2  36689
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