Theorem difid 3489

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 3173	. 2 ⊢ 𝐴 ⊆ 𝐴
2		ssdif0im 3485	. 2 ⊢ (𝐴 ⊆ 𝐴 → (𝐴 ∖ 𝐴) = ∅)
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ∖ 𝐴) = ∅

Syntax hints:   = wceq 1353   ∖ cdif 3124   ⊆ wss 3127  ∅c0 3420

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-v 2737  df-dif 3129  df-in 3133  df-ss 3140  df-nul 3421

This theorem is referenced by:  dif0  3491  difun2  3500  diftpsn3  3730  2oconcl  6430  ismkvnex  7143  topcld  13180