Theorem wunelss 9845

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 9842	. . 3 ⊢ (𝑈 ∈ WUni → Tr 𝑈)
3	1, 2	syl 17	. 2 ⊢ (𝜑 → Tr 𝑈)
4		wununi.2	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
5		trss 4984	. 2 ⊢ (Tr 𝑈 → (𝐴 ∈ 𝑈 → 𝐴 ⊆ 𝑈))
6	3, 4, 5	sylc 65	1 ⊢ (𝜑 → 𝐴 ⊆ 𝑈)

Syntax hints:   → wi 4   ∈ wcel 2164   ⊆ wss 3798  Tr wtr 4975  WUnicwun 9837

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-ext 2803

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-v 3416  df-in 3805  df-ss 3812  df-uni 4659  df-tr 4976  df-wun 9839

This theorem is referenced by:  wunss  9849  wunf  9864  wuncval2  9884