Theorem wuntr 10660

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 10659	. . 3 ⊢ (𝑈 ∈ WUni → (𝑈 ∈ WUni ↔ (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈))))
2	1	ibi 269	. 2 ⊢ (𝑈 ∈ WUni → (Tr 𝑈 ∧ 𝑈 ≠ ∅ ∧ ∀𝑥 ∈ 𝑈 (∪ 𝑥 ∈ 𝑈 ∧ 𝒫 𝑥 ∈ 𝑈 ∧ ∀𝑦 ∈ 𝑈 {𝑥, 𝑦} ∈ 𝑈)))
3	2	simp1d 1154	1 ⊢ (𝑈 ∈ WUni → Tr 𝑈)

Syntax hints:   → wi 4   ∧ w3a 1097   ∈ wcel 2141   ≠ wne 2956  ∀wral 3075  ∅c0 4285  𝒫 cpw 4554  {cpr 4583  ∪ cuni 4864  Tr wtr 5206  WUnicwun 10655

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-3an 1099  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rex 3086  df-v 3455  df-ss 3921  df-uni 4865  df-tr 5207  df-wun 10657

This theorem is referenced by:  wunelss  10663  intwun  10690