Theorem iswun 10770

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 5219	. . 3 ⊢ (𝑢 = 𝑈 → (Tr 𝑢 ↔ Tr 𝑈))
2		neeq1 3018	. . 3 ⊢ (𝑢 = 𝑈 → (𝑢 ≠ ∅ ↔ 𝑈 ≠ ∅))
3		eleq2 2850	. . . . 5 ⊢ (𝑢 = 𝑈 → (∪ 𝑥 ∈ 𝑢 ↔ ∪ 𝑥 ∈ 𝑈))
4		eleq2 2850	. . . . 5 ⊢ (𝑢 = 𝑈 → (𝒫 𝑥 ∈ 𝑢 ↔ 𝒫 𝑥 ∈ 𝑈))
5		eleq2 2850	. . . . . 6 ⊢ (𝑢 = 𝑈 → ({𝑥, 𝑦} ∈ 𝑢 ↔ {𝑥, 𝑦} ∈ 𝑈))
6	5	raleqbi1dv 3330	. . . . 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 3330	. . 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 10768	. 2 ⊢ WUni = {𝑢 ∣ (Tr 𝑢 ∧ 𝑢 ≠ ∅ ∧ ∀𝑥 ∈ 𝑢 (∪ 𝑥 ∈ 𝑢 ∧ 𝒫 𝑥 ∈ 𝑢 ∧ ∀𝑦 ∈ 𝑢 {𝑥, 𝑦} ∈ 𝑢))}
11	9, 10	elab2g 3634	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 1570   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077  ∅c0 4279  𝒫 cpw 4557  {cpr 4586  ∪ cuni 4867  Tr wtr 5212  WUnicwun 10766

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-v 3453  df-ss 3916  df-uni 4868  df-tr 5213  df-wun 10768

This theorem is used by:  wuntr  10771  wununi  10772  wunpw  10773  wunpr  10775  wun0  10784  intwun  10801  r1limwun  10802  wunex2  10804  tskwun  10850  gruwun  10879  pwinfi2  44521