Theorem difidALT 3949

Description: Alternate proof of difid 3948. (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 3583	. 2 ⊢ (𝐴 ∖ 𝐴) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐴}
2		dfnul3 3918	. 2 ⊢ ∅ = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐴}
3	1, 2	eqtr4i 2647	1 ⊢ (𝐴 ∖ 𝐴) = ∅

Syntax hints:  ¬ wn 3   = wceq 1483   ∈ wcel 1990  {crab 2916   ∖ cdif 3571  ∅c0 3915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-nul 3916

This theorem is referenced by: (None)