Theorem wunelss 10599

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 10596	. . 3 ⊢ (𝑈 ∈ WUni → Tr 𝑈)
3	1, 2	syl 17	. 2 ⊢ (𝜑 → Tr 𝑈)
4		wununi.2	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
5		trss 5206	. 2 ⊢ (Tr 𝑈 → (𝐴 ∈ 𝑈 → 𝐴 ⊆ 𝑈))
6	3, 4, 5	sylc 65	1 ⊢ (𝜑 → 𝐴 ⊆ 𝑈)

Syntax hints:   → wi 4   ∈ wcel 2111   ⊆ wss 3897  Tr wtr 5196  WUnicwun 10591

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 2113  ax-9 2121  ax-ext 2703

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 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-v 3438  df-ss 3914  df-uni 4857  df-tr 5197  df-wun 10593

This theorem is referenced by:  wunss  10603  wunf  10618  wuncval2  10638