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Theorem 0in 3449
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  |-  ( (/)  i^i 
A )  =  (/)

Proof of Theorem 0in
StepHypRef Expression
1 incom 3319 . 2  |-  ( (/)  i^i 
A )  =  ( A  i^i  (/) )
2 in0 3448 . 2  |-  ( A  i^i  (/) )  =  (/)
31, 2eqtri 2191 1  |-  ( (/)  i^i 
A )  =  (/)
Colors of variables: wff set class
Syntax hints:    = wceq 1348    i^i cin 3120   (/)c0 3414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-in 3127  df-nul 3415
This theorem is referenced by:  setsfun  12444  setsfun0  12445  restsn  12939
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