Theorem 0in 3444

Description: The intersection of the empty set with a class is the empty set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Assertion

Ref	Expression
0in	⊢ (∅ ∩ 𝐴) = ∅

Proof of Theorem 0in

Step	Hyp	Ref	Expression
1		incom 3314	. 2 ⊢ (∅ ∩ 𝐴) = (𝐴 ∩ ∅)
2		in0 3443	. 2 ⊢ (𝐴 ∩ ∅) = ∅
3	1, 2	eqtri 2186	1 ⊢ (∅ ∩ 𝐴) = ∅

Syntax hints:   = wceq 1343   ∩ cin 3115  ∅c0 3409

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-dif 3118  df-in 3122  df-nul 3410

This theorem is referenced by:  setsfun  12429  setsfun0  12430  restsn  12820