Theorem difidALT 4342

Description: Alternate proof of difid 4341. Shorter, but requiring ax-8 2111, df-clel 2804. (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 3971	. 2 ⊢ 𝐴 ⊆ 𝐴
2		ssdif0 4331	. 2 ⊢ (𝐴 ⊆ 𝐴 ↔ (𝐴 ∖ 𝐴) = ∅)
3	1, 2	mpbi 230	1 ⊢ (𝐴 ∖ 𝐴) = ∅

Syntax hints:   = wceq 1540   ∖ cdif 3913   ⊆ wss 3916  ∅c0 4298

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3452  df-dif 3919  df-ss 3933  df-nul 4299

This theorem is referenced by: (None)