Theorem 0in 3486

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 3355	. 2 ⊢ (∅ ∩ 𝐴) = (𝐴 ∩ ∅)
2		in0 3485	. 2 ⊢ (𝐴 ∩ ∅) = ∅
3	1, 2	eqtri 2217	1 ⊢ (∅ ∩ 𝐴) = ∅

Syntax hints:   = wceq 1364   ∩ cin 3156  ∅c0 3450

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-dif 3159  df-in 3163  df-nul 3451

This theorem is referenced by:  setsfun  12713  setsfun0  12714  restsn  14416