Theorem 0in 4350

Description: The intersection of the empty set with a class is the empty set. Commuted form of 0in 4350. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Assertion

Ref	Expression
0in	⊢ (∅ ∩ 𝐴) = ∅

Proof of Theorem 0in

Step	Hyp	Ref	Expression
1		in0 4348	. 2 ⊢ (𝐴 ∩ ∅) = ∅
2	1	ineqcomi 4111	1 ⊢ (∅ ∩ 𝐴) = ∅

This proof depends on syntax axioms:   = wceq 1570   ∩ cin 4092  ∅c0 4281

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-dif 4050  df-in 4094  df-nul 4282

This theorem is used by:  pred0  6359  fresaunres2  6779  fnsuppeq0  8240  setsfun  17381  setsfun0  17382  indistopon  23352  fctop  23355  cctop  23357  restsn  23521  filconn  24235  chtdif  27518  ppidif  27523  ppi1  27524  cht1  27525  0res  33272  ofpreima2  33335  ordtconnlem1  34639  measvuni  34930  measinb  34937  cndprobnul  35152  ballotlemfp1  35207  ballotlemgun  35240  chtvalz  35341  mrsubvrs  36467  mblfinlem2  38781  ntrkbimka  45241  neicvgbex  45315  limsup0  46919  subsalsal  47584  nnfoctbdjlem  47680  setc1onsubc  50955