Theorem 0in 3398

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 3268	. 2 ⊢ (∅ ∩ 𝐴) = (𝐴 ∩ ∅)
2		in0 3397	. 2 ⊢ (𝐴 ∩ ∅) = ∅
3	1, 2	eqtri 2160	1 ⊢ (∅ ∩ 𝐴) = ∅

Syntax hints:   = wceq 1331   ∩ cin 3070  ∅c0 3363

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-dif 3073  df-in 3077  df-nul 3364

This theorem is referenced by:  setsfun  11994  setsfun0  11995  restsn  12349