Theorem wuntr 10771

Description: A weak universe is transitive. (Contributed by Mario Carneiro, 2-Jan-2017.)

Assertion

Ref	Expression
wuntr	⊢ (𝑈 ∈ WUni → Tr 𝑈)

Proof of Theorem wuntr

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iswun 10770	. . 3 ⊢ (𝑈 ∈ WUni → (𝑈 ∈ WUni ↔ (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈))))
2	1	ibi 270	. 2 ⊢ (𝑈 ∈ WUni → (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈)))
3	2	simp1d 1160	1 ⊢ (𝑈 ∈ WUni → Tr 𝑈)

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   ∈ 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:  wunelss  10774  intwun  10801