Theorem wunelss 10685

Description: The elements of a weak universe are also subsets of it. (Contributed by Mario Carneiro, 2-Jan-2017.)

Hypotheses

Ref	Expression
wununi.1	⊢ (𝜑 → 𝑈 ∈ WUni)
wununi.2	⊢ (𝜑 → 𝐴 ∈ 𝑈)

Assertion

Ref	Expression
wunelss	⊢ (𝜑 → 𝐴 ⊆ 𝑈)

Proof of Theorem wunelss

Step	Hyp	Ref	Expression
1		wununi.1	. . 3 ⊢ (𝜑 → 𝑈 ∈ WUni)
2		wuntr 10682	. . 3 ⊢ (𝑈 ∈ WUni → Tr 𝑈)
3	1, 2	syl 17	. 2 ⊢ (𝜑 → Tr 𝑈)
4		wununi.2	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
5		trss 5269	. 2 ⊢ (Tr 𝑈 → (𝐴 ∈ 𝑈 → 𝐴 ⊆ 𝑈))
6	3, 4, 5	sylc 65	1 ⊢ (𝜑 → 𝐴 ⊆ 𝑈)

Syntax hints:   → wi 4   ∈ wcel 2106   ⊆ wss 3944  Tr wtr 5258  WUnicwun 10677

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1089  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-v 3475  df-in 3951  df-ss 3961  df-uni 4902  df-tr 5259  df-wun 10679

This theorem is referenced by:  wunss  10689  wunf  10704  wuncval2  10724