Theorem difidALT 4305

Description: Alternate proof of difid 4304. Shorter, but requiring ax-8 2108, df-clel 2816. (Contributed by NM, 22-Apr-2004.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
difidALT	⊢ (𝐴 ∖ 𝐴) = ∅

Proof of Theorem difidALT

Step	Hyp	Ref	Expression
1		ssid 3943	. 2 ⊢ 𝐴 ⊆ 𝐴
2		ssdif0 4297	. 2 ⊢ (𝐴 ⊆ 𝐴 ↔ (𝐴 ∖ 𝐴) = ∅)
3	1, 2	mpbi 229	1 ⊢ (𝐴 ∖ 𝐴) = ∅

Syntax hints:   = wceq 1539   ∖ cdif 3884   ⊆ wss 3887  ∅c0 4256

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-dif 3890  df-in 3894  df-ss 3904  df-nul 4257

This theorem is referenced by: (None)