Theorem 0in 4337

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		in0 4335	. 2 ⊢ (𝐴 ∩ ∅) = ∅
2	1	ineqcomi 4151	1 ⊢ (∅ ∩ 𝐴) = ∅

Syntax hints:   = wceq 1542   ∩ cin 3888  ∅c0 4273

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3390  df-v 3431  df-dif 3892  df-in 3896  df-nul 4274

This theorem is referenced by:  pred0  6299  fresaunres2  6712  fnsuppeq0  8142  setsfun  17141  setsfun0  17142  indistopon  22966  fctop  22969  cctop  22971  restsn  23135  filconn  23848  chtdif  27121  ppidif  27126  ppi1  27127  cht1  27128  0res  32673  ofpreima2  32739  ordtconnlem1  34068  measvuni  34358  measinb  34365  cndprobnul  34581  ballotlemfp1  34636  ballotlemgun  34669  chtvalz  34773  mrsubvrs  35704  mblfinlem2  37979  ntrkbimka  44465  neicvgbex  44539  limsup0  46122  subsalsal  46787  nnfoctbdjlem  46883  setc1onsubc  50077