Theorem difidALT 4286

Description: Alternate proof of difid 4285. Shorter, but requiring ax-8 2112, 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 3923	. 2 ⊢ 𝐴 ⊆ 𝐴
2		ssdif0 4278	. 2 ⊢ (𝐴 ⊆ 𝐴 ↔ (𝐴 ∖ 𝐴) = ∅)
3	1, 2	mpbi 233	1 ⊢ (𝐴 ∖ 𝐴) = ∅

Syntax hints:   = wceq 1543   ∖ cdif 3863   ⊆ wss 3866  ∅c0 4237

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3410  df-dif 3869  df-in 3873  df-ss 3883  df-nul 4238

This theorem is referenced by: (None)