Theorem wundif 10783

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 4160	. 2 ⊢ (𝜑 → (𝐴 ∖ 𝐵) ⊆ 𝐴)
4	1, 2, 3	wunss 10781	1 ⊢ (𝜑 → (𝐴 ∖ 𝐵) ∈ 𝑈)

Syntax hints:   → wi 4   ∈ wcel 2108   ∖ cdif 3973  WUnicwun 10769

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-in 3983  df-ss 3993  df-pw 4624  df-uni 4932  df-tr 5284  df-wun 10771

This theorem is referenced by:  wuncn  11239