Theorem difidALT 4329

Description: Alternate proof of difid 4328. Shorter, but requiring ax-8 2147, df-clel 2837. (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 3956	. 2 ⊢ 𝐴 ⊆ 𝐴
2		ssdif0 4317	. 2 ⊢ (𝐴 ⊆ 𝐴 ↔ (𝐴 ∖ 𝐴) = ∅)
3	1, 2	mpbi 233	1 ⊢ (𝐴 ∖ 𝐴) = ∅

This proof depends on syntax axioms:   = wceq 1570   ∖ cdif 3899   ⊆ wss 3902  ∅c0 4282

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3455  df-dif 3905  df-ss 3919  df-nul 4283

This theorem is used by: (None)