Theorem dif0 3399

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 3397	. . 3 ⊢ (𝐴 ∖ 𝐴) = ∅
2	1	difeq2i 3157	. 2 ⊢ (𝐴 ∖ (𝐴 ∖ 𝐴)) = (𝐴 ∖ ∅)
3		difdif 3167	. 2 ⊢ (𝐴 ∖ (𝐴 ∖ 𝐴)) = 𝐴
4	2, 3	eqtr3i 2137	1 ⊢ (𝐴 ∖ ∅) = 𝐴

Syntax hints:   = wceq 1314   ∖ cdif 3034  ∅c0 3329

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rab 2399  df-v 2659  df-dif 3039  df-in 3043  df-ss 3050  df-nul 3330

This theorem is referenced by:  disjdif2  3407  2oconcl  6290  diffifi  6741  undifdc  6765  difinfinf  6938  ismkvnex  6979  0cld  12124  exmid1stab  12887