Theorem 0in 3532

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 3401	. 2 ⊢ (∅ ∩ 𝐴) = (𝐴 ∩ ∅)
2		in0 3531	. 2 ⊢ (𝐴 ∩ ∅) = ∅
3	1, 2	eqtri 2252	1 ⊢ (∅ ∩ 𝐴) = ∅

Syntax hints:   = wceq 1398   ∩ cin 3200  ∅c0 3496

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-dif 3203  df-in 3207  df-nul 3497

This theorem is referenced by:  setsfun  13178  setsfun0  13179  restsn  14971