Theorem wuntr 10606

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 10605	. . 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 1142	1 ⊢ (𝑈 ∈ WUni → Tr 𝑈)

Syntax hints:   → wi 4   ∧ w3a 1086   ∈ wcel 2113   ≠ wne 2930  ∀wral 3049  ∅c0 4284  𝒫 cpw 4551  {cpr 4579  ∪ cuni 4860  Tr wtr 5202  WUnicwun 10601

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2931  df-ral 3050  df-rex 3059  df-v 3440  df-ss 3916  df-uni 4861  df-tr 5203  df-wun 10603

This theorem is referenced by:  wunelss  10609  intwun  10636