Theorem difidALT 4308

Description: Alternate proof of difid 4307. Shorter, but requiring ax-8 2123, 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 3939	. 2 ⊢ 𝐴 ⊆ 𝐴
2		ssdif0 4297	. 2 ⊢ (𝐴 ⊆ 𝐴 ↔ (𝐴 ∖ 𝐴) = ∅)
3	1, 2	mpbi 232	1 ⊢ (𝐴 ∖ 𝐴) = ∅

Syntax hints:   = wceq 1548   ∖ cdif 3882   ⊆ wss 3885  ∅c0 4264

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 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-v 3435  df-dif 3888  df-ss 3902  df-nul 4265

This theorem is referenced by: (None)