Theorem difid 3431

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 3117	. 2 ⊢ 𝐴 ⊆ 𝐴
2		ssdif0im 3427	. 2 ⊢ (𝐴 ⊆ 𝐴 → (𝐴 ∖ 𝐴) = ∅)
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∖ 𝐴) = ∅

Syntax hints:   = wceq 1331   ∖ cdif 3068   ⊆ wss 3071  ∅c0 3363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-dif 3073  df-in 3077  df-ss 3084  df-nul 3364

This theorem is referenced by:  dif0  3433  difun2  3442  diftpsn3  3661  2oconcl  6336  ismkvnex  7029  topcld  12278