Theorem difid 3515

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 3199	. 2 ⊢ 𝐴 ⊆ 𝐴
2		ssdif0im 3511	. 2 ⊢ (𝐴 ⊆ 𝐴 → (𝐴 ∖ 𝐴) = ∅)
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∖ 𝐴) = ∅

Syntax hints:   = wceq 1364   ∖ cdif 3150   ⊆ wss 3153  ∅c0 3446

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-dif 3155  df-in 3159  df-ss 3166  df-nul 3447

This theorem is referenced by:  dif0  3517  difun2  3526  diftpsn3  3759  2oconcl  6492  ismkvnex  7214  topcld  14277