Theorem iswun 10792

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 5218	. . 3 ⊢ (𝑢 = 𝑈 → (Tr 𝑢 ↔ Tr 𝑈))
2		neeq1 3017	. . 3 ⊢ (𝑢 = 𝑈 → (𝑢 ≠ ∅ ↔ 𝑈 ≠ ∅))
3		eleq2 2849	. . . . 5 ⊢ (𝑢 = 𝑈 → (∪ 𝑥 ∈ 𝑢 ↔ ∪ 𝑥 ∈ 𝑈))
4		eleq2 2849	. . . . 5 ⊢ (𝑢 = 𝑈 → (𝒫 𝑥 ∈ 𝑢 ↔ 𝒫 𝑥 ∈ 𝑈))
5		eleq2 2849	. . . . . 6 ⊢ (𝑢 = 𝑈 → ({𝑥, 𝑦} ∈ 𝑢 ↔ {𝑥, 𝑦} ∈ 𝑈))
6	5	raleqbi1dv 3329	. . . . 5 ⊢ (𝑢 = 𝑈 → (∀𝑦 ∈ 𝑢 {𝑥, 𝑦} ∈ 𝑢 ↔ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈))
7	3, 4, 6	3anbi123d 1464	. . . 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 3329	. . 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 1464	. 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 10790	. 2 ⊢ WUni = {𝑢 ∣ (Tr 𝑢 ∧ 𝑢 ≠ ∅ ∧ ∀𝑥 ∈ 𝑢 (∪ 𝑥 ∈ 𝑢 ∧ 𝒫 𝑥 ∈ 𝑢 ∧ ∀𝑦 ∈ 𝑢 {𝑥, 𝑦} ∈ 𝑢))}
11	9, 10	elab2g 3633	1 ⊢ (𝑈 ∈ 𝑉 → (𝑈 ∈ WUni ↔ (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  ∅c0 4278  𝒫 cpw 4556  {cpr 4586  ∪ cuni 4866  Tr wtr 5211  WUnicwun 10788

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-v 3452  df-ss 3896  df-uni 4867  df-tr 5212  df-wun 10790

This theorem is used by:  wuntr  10793  wununi  10794  wunpw  10795  wunpr  10797  wun0  10806  intwun  10823  r1limwun  10824  wunex2  10826  tskwun  10872  gruwun  10901  pwinfi2  44656