Theorem difidALT 4330

Description: Alternate proof of difid 4329. (Contributed by David Abernethy, 17-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
difidALT	⊢ (𝐴 ∖ 𝐴) = ∅

Proof of Theorem difidALT

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfdif2 3944	. 2 ⊢ (𝐴 ∖ 𝐴) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐴}
2		dfnul3 4294	. 2 ⊢ ∅ = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐴}
3	1, 2	eqtr4i 2847	1 ⊢ (𝐴 ∖ 𝐴) = ∅

Syntax hints:  ¬ wn 3   = wceq 1533   ∈ wcel 2110  {crab 3142   ∖ cdif 3932  ∅c0 4290

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-9 2120  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1777  df-sb 2066  df-clab 2800  df-cleq 2814  df-rab 3147  df-dif 3938  df-nul 4291

This theorem is referenced by: (None)