Theorem difidALT 3620

Description: The difference between a class and itself is the empty set. Proposition 5.15 of [TakeutiZaring] p. 20. Also Theorem 32 of [Suppes] p. 28. Alternate proof of difid 3619. (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 3223	. 2 ⊢ (A ∖ A) = {x ∈ A ∣ ¬ x ∈ A}
2		dfnul3 3554	. 2 ⊢ ∅ = {x ∈ A ∣ ¬ x ∈ A}
3	1, 2	eqtr4i 2376	1 ⊢ (A ∖ A) = ∅

Syntax hints:  ¬ wn 3   = wceq 1642   ∈ wcel 1710  {crab 2619   ∖ cdif 3207  ∅c0 3551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-rab 2624  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-nul 3552

This theorem is referenced by: (None)