Theorem wundif 10672

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 4090	. 2 ⊢ (𝜑 → (𝐴 ∖ 𝐵) ⊆ 𝐴)
4	1, 2, 3	wunss 10670	1 ⊢ (𝜑 → (𝐴 ∖ 𝐵) ∈ 𝑈)

Syntax hints:   → wi 4   ∈ wcel 2142   ∖ cdif 3901  WUnicwun 10658

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5246

This theorem depends on definitions:  df-bi 209  df-an 400  df-3an 1100  df-tru 1563  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-in 3911  df-ss 3921  df-pw 4557  df-uni 4866  df-tr 5208  df-wun 10660

This theorem is referenced by:  wuncn  11128