Theorem difidALT 4370

Description: Alternate proof of difid 4369. Shorter, but requiring ax-8 2108, df-clel 2810. (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 4003	. 2 ⊢ 𝐴 ⊆ 𝐴
2		ssdif0 4362	. 2 ⊢ (𝐴 ⊆ 𝐴 ↔ (𝐴 ∖ 𝐴) = ∅)
3	1, 2	mpbi 229	1 ⊢ (𝐴 ∖ 𝐴) = ∅

Syntax hints:   = wceq 1541   ∖ cdif 3944   ⊆ wss 3947  ∅c0 4321

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-dif 3950  df-in 3954  df-ss 3964  df-nul 4322

This theorem is referenced by: (None)