Theorem wuntr 10745

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 10744	. . 3 ⊢ (𝑈 ∈ WUni → (𝑈 ∈ WUni ↔ (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈))))
2	1	ibi 267	. 2 ⊢ (𝑈 ∈ WUni → (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈)))
3	2	simp1d 1143	1 ⊢ (𝑈 ∈ WUni → Tr 𝑈)

Syntax hints:   → wi 4   ∧ w3a 1087   ∈ wcel 2108   ≠ wne 2940  ∀wral 3061  ∅c0 4333  𝒫 cpw 4600  {cpr 4628  ∪ cuni 4907  Tr wtr 5259  WUnicwun 10740

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-v 3482  df-ss 3968  df-uni 4908  df-tr 5260  df-wun 10742

This theorem is referenced by:  wunelss  10748  intwun  10775