Theorem difid 3581

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. (Contributed by NM, 22-Apr-2004.)

Assertion

Ref	Expression
difid	⊢ (𝐴 ∖ 𝐴) = ∅

Proof of Theorem difid

Step	Hyp	Ref	Expression
1		ssid 3262	. 2 ⊢ 𝐴 ⊆ 𝐴
2		ssdif0im 3577	. 2 ⊢ (𝐴 ⊆ 𝐴 → (𝐴 ∖ 𝐴) = ∅)
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∖ 𝐴) = ∅

Syntax hints:   = wceq 1398   ∖ cdif 3211   ⊆ wss 3214  ∅c0 3512

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-dif 3216  df-in 3220  df-ss 3227  df-nul 3513

This theorem is referenced by:  dif0  3583  difun2  3593  diftpsn3  3840  2oconcl  6685  ismkvnex  7459  topcld  15086