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Theorem strsl0 12044
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 4062 . . 3  |-  (/)  e.  _V
2 strsl0.e . . . 4  |-  ( E  = Slot  ( E `  ndx )  /\  ( E `  ndx )  e.  NN )
32simpli 110 . . 3  |-  E  = Slot  ( E `  ndx )
42simpri 112 . . 3  |-  ( E `
 ndx )  e.  NN
51, 3, 4strnfvn 12017 . 2  |-  ( E `
 (/) )  =  (
(/) `  ( E `  ndx ) )
6 0fv 5463 . 2  |-  ( (/) `  ( E `  ndx ) )  =  (/)
75, 6eqtr2i 2162 1  |-  (/)  =  ( E `  (/) )
Colors of variables: wff set class
Syntax hints:    /\ wa 103    = wceq 1332    e. wcel 1481   (/)c0 3367   ` cfv 5130   NNcn 8743   ndxcnx 11993  Slot cslot 11995
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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138  ax-un 4362
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-sbc 2913  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-iota 5095  df-fun 5132  df-fv 5138  df-slot 12000
This theorem is referenced by:  base0  12045
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