ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  strsl0 Unicode version

Theorem strsl0 12510
Description: All components of the empty set are empty sets. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Jim Kingdon, 31-Jan-2023.)
Hypothesis
Ref Expression
strsl0.e  |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )
Assertion
Ref Expression
strsl0  |-  (/)  =  ( E `  (/) )

Proof of Theorem strsl0
StepHypRef Expression
1 0ex 4130 . . 3  |-  (/)  e.  _V
2 strsl0.e . . . 4  |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )
32simpli 111 . . 3  |-  E  = Slot  ( E `  ndx )
42simpri 113 . . 3  |-  ( E `
 ndx )  e.  NN
51, 3, 4strnfvn 12482 . 2  |-  ( E `
 (/) )  =  (
(/) `  ( E `  ndx ) )
6 0fv 5550 . 2  |-  ( (/) `  ( E `  ndx ) )  =  (/)
75, 6eqtr2i 2199 1  |-  (/)  =  ( E `  (/) )
Colors of variables: wff set class
Syntax hints:    /\ wa 104    = wceq 1353    e. wcel 2148   (/)c0 3422   ` cfv 5216   NNcn 8918   ndxcnx 12458  Slot cslot 12460
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-iota 5178  df-fun 5218  df-fv 5224  df-slot 12465
This theorem is referenced by:  base0  12511
  Copyright terms: Public domain W3C validator