Theorem difidALT 4386

Description: Alternate proof of difid 4385. Shorter, but requiring ax-8 2110, 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 4021	. 2 ⊢ 𝐴 ⊆ 𝐴
2		ssdif0 4375	. 2 ⊢ (𝐴 ⊆ 𝐴 ↔ (𝐴 ∖ 𝐴) = ∅)
3	1, 2	mpbi 230	1 ⊢ (𝐴 ∖ 𝐴) = ∅

Syntax hints:   = wceq 1539   ∖ cdif 3963   ⊆ wss 3966  ∅c0 4342

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3483  df-dif 3969  df-ss 3983  df-nul 4343

This theorem is referenced by: (None)