Theorem wunelss 10720

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 10717	. . 3 ⊢ (𝑈 ∈ WUni → Tr 𝑈)
3	1, 2	syl 18	. 2 ⊢ (𝜑 → Tr 𝑈)
4		wununi.2	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
5		trss 5222	. 2 ⊢ (Tr 𝑈 → (𝐴 ∈ 𝑈 → 𝐴 ⊆ 𝑈))
6	3, 4, 5	sylc 66	1 ⊢ (𝜑 → 𝐴 ⊆ 𝑈)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145   ⊆ wss 3899  Tr wtr 5212  WUnicwun 10712

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-v 3452  df-ss 3916  df-uni 4868  df-tr 5213  df-wun 10714

This theorem is used by:  wunss  10724  wunf  10739  wuncval2  10759