Theorem wundif 10694

Description: A weak universe is closed under class difference. (Contributed by Mario Carneiro, 2-Jan-2017.)

Hypotheses

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

Assertion

Ref	Expression
wundif	⊢ (𝜑 → (𝐴 ∖ 𝐵) ∈ 𝑈)

Proof of Theorem wundif

Step	Hyp	Ref	Expression
1		wununi.1	. 2 ⊢ (𝜑 → 𝑈 ∈ WUni)
2		wununi.2	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
3		difssd 4091	. 2 ⊢ (𝜑 → (𝐴 ∖ 𝐵) ⊆ 𝐴)
4	1, 2, 3	wunss 10692	1 ⊢ (𝜑 → (𝐴 ∖ 𝐵) ∈ 𝑈)

Syntax hints:   → wi 4   ∈ wcel 2143   ∖ cdif 3902  WUnicwun 10680

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257

This theorem depends on definitions:  df-bi 210  df-an 401  df-3an 1105  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-in 3912  df-ss 3922  df-pw 4564  df-uni 4873  df-tr 5219  df-wun 10682

This theorem is referenced by:  wuncn  11150