Theorem 0in 4346

Description: The intersection of the empty set with a class is the empty set. Commuted form of 0in 4346. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Assertion

Ref	Expression
0in	⊢ (∅ ∩ 𝐴) = ∅

Proof of Theorem 0in

Step	Hyp	Ref	Expression
1		in0 4344	. 2 ⊢ (𝐴 ∩ ∅) = ∅
2	1	ineqcomi 4108	1 ⊢ (∅ ∩ 𝐴) = ∅

This proof depends on syntax axioms:   = wceq 1569   ∩ cin 4089  ∅c0 4278

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 4047  df-in 4091  df-nul 4279

This theorem is used by:  pred0  6338  fresaunres2  6755  fnsuppeq0  8213  setsfun  17355  setsfun0  17356  indistopon  23325  fctop  23328  cctop  23330  restsn  23494  filconn  24208  chtdif  27491  ppidif  27496  ppi1  27497  cht1  27498  0res  33245  ofpreima2  33308  ordtconnlem1  34603  measvuni  34894  measinb  34901  cndprobnul  35116  ballotlemfp1  35171  ballotlemgun  35204  chtvalz  35305  mrsubvrs  36374  mblfinlem2  38670  ntrkbimka  45132  neicvgbex  45206  limsup0  46798  subsalsal  47463  nnfoctbdjlem  47559  setc1onsubc  50834