Theorem dif0 3578

Description: The difference between a class and the empty set. Part of Exercise 4.4 of [Stoll] p. 16. (Contributed by NM, 17-Aug-2004.)

Assertion

Ref	Expression
dif0	⊢ (𝐴 ∖ ∅) = 𝐴

Proof of Theorem dif0

Step	Hyp	Ref	Expression
1		difid 3576	. . 3 ⊢ (𝐴 ∖ 𝐴) = ∅
2	1	difeq2i 3333	. 2 ⊢ (𝐴 ∖ (𝐴 ∖ 𝐴)) = (𝐴 ∖ ∅)
3		difdif 3343	. 2 ⊢ (𝐴 ∖ (𝐴 ∖ 𝐴)) = 𝐴
4	2, 3	eqtr3i 2255	1 ⊢ (𝐴 ∖ ∅) = 𝐴

Syntax hints:   = wceq 1398   ∖ cdif 3207  ∅c0 3507

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 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rab 2529  df-v 2814  df-dif 3212  df-in 3216  df-ss 3223  df-nul 3508

This theorem is referenced by:  disjdif2  3587  exmid1stab  4320  2oconcl  6671  diffifi  7150  undifdc  7183  difinfinf  7391  ismkvnex  7445  m1bits  12642  0cld  14969