Theorem 0in 3450

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 3319	. 2 ⊢ (∅ ∩ 𝐴) = (𝐴 ∩ ∅)
2		in0 3449	. 2 ⊢ (𝐴 ∩ ∅) = ∅
3	1, 2	eqtri 2191	1 ⊢ (∅ ∩ 𝐴) = ∅

Syntax hints:   = wceq 1348   ∩ cin 3120  ∅c0 3414

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-in 3127  df-nul 3415

This theorem is referenced by:  setsfun  12451  setsfun0  12452  restsn  12974