Theorem iswun 10627

Description: Properties of a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.)

Assertion

Ref	Expression
iswun	⊢ (𝑈 ∈ 𝑉 → (𝑈 ∈ WUni ↔ (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {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.

Step	Hyp	Ref	Expression
1		treq 5214	. . 3 ⊢ (𝑢 = 𝑈 → (Tr 𝑢 ↔ Tr 𝑈))
2		neeq1 2995	. . 3 ⊢ (𝑢 = 𝑈 → (𝑢 ≠ ∅ ↔ 𝑈 ≠ ∅))
3		eleq2 2826	. . . . 5 ⊢ (𝑢 = 𝑈 → (∪ 𝑥 ∈ 𝑢 ↔ ∪ 𝑥 ∈ 𝑈))
4		eleq2 2826	. . . . 5 ⊢ (𝑢 = 𝑈 → (𝒫 𝑥 ∈ 𝑢 ↔ 𝒫 𝑥 ∈ 𝑈))
5		eleq2 2826	. . . . . 6 ⊢ (𝑢 = 𝑈 → ({𝑥, 𝑦} ∈ 𝑢 ↔ {𝑥, 𝑦} ∈ 𝑈))
6	5	raleqbi1dv 3310	. . . . 5 ⊢ (𝑢 = 𝑈 → (∀𝑦 ∈ 𝑢 {𝑥, 𝑦} ∈ 𝑢 ↔ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈))
7	3, 4, 6	3anbi123d 1439	. . . 4 ⊢ (𝑢 = 𝑈 → ((∪ 𝑥 ∈ 𝑢 ∧ 𝒫 𝑥 ∈ 𝑢 ∧ ∀𝑦 ∈ 𝑢 {𝑥, 𝑦} ∈ 𝑢) ↔ (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈)))
8	7	raleqbi1dv 3310	. . 3 ⊢ (𝑢 = 𝑈 → (∀𝑥 ∈ 𝑢 (∪ 𝑥 ∈ 𝑢 ∧ 𝒫 𝑥 ∈ 𝑢 ∧ ∀𝑦 ∈ 𝑢 {𝑥, 𝑦} ∈ 𝑢) ↔ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈)))
9	1, 2, 8	3anbi123d 1439	. 2 ⊢ (𝑢 = 𝑈 → ((Tr 𝑢 ∧ 𝑢 ≠ ∅ ∧ ∀𝑥 ∈ 𝑢 (∪ 𝑥 ∈ 𝑢 ∧ 𝒫 𝑥 ∈ 𝑢 ∧ ∀𝑦 ∈ 𝑢 {𝑥, 𝑦} ∈ 𝑢)) ↔ (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈))))
10		df-wun 10625	. 2 ⊢ WUni = {𝑢 ∣ (Tr 𝑢 ∧ 𝑢 ≠ ∅ ∧ ∀𝑥 ∈ 𝑢 (∪ 𝑥 ∈ 𝑢 ∧ 𝒫 𝑥 ∈ 𝑢 ∧ ∀𝑦 ∈ 𝑢 {𝑥, 𝑦} ∈ 𝑢))}
11	9, 10	elab2g 3637	1 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∈ WUni ↔ (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∅c0 4287  𝒫 cpw 4556  {cpr 4584  ∪ cuni 4865  Tr wtr 5207  WUnicwun 10623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-v 3444  df-ss 3920  df-uni 4866  df-tr 5208  df-wun 10625

This theorem is referenced by:  wuntr  10628  wununi  10629  wunpw  10630  wunpr  10632  wun0  10641  intwun  10658  r1limwun  10659  wunex2  10661  tskwun  10707  gruwun  10736  pwinfi2  43912