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

Theorem basendxnn 12487
Description: The index value of the base set extractor is a positive integer. This property should be ensured for every concrete coding because otherwise it could not be used in an extensible structure (slots must be positive integers). (Contributed by AV, 23-Sep-2020.)
Assertion
Ref Expression
basendxnn (Base‘ndx) ∈ ℕ

Proof of Theorem basendxnn
StepHypRef Expression
1 df-base 12438 . . 3 Base = Slot 1
2 1nn 8906 . . 3 1 ∈ ℕ
31, 2ndxarg 12455 . 2 (Base‘ndx) = 1
43, 2eqeltri 2250 1 (Base‘ndx) ∈ ℕ
Colors of variables: wff set class
Syntax hints:  wcel 2148  cfv 5211  1c1 7790  cn 8895  ndxcnx 12429  Basecbs 12432
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-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 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-cnex 7880  ax-resscn 7881  ax-1re 7883  ax-addrcl 7886
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  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-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-iota 5173  df-fun 5213  df-fv 5219  df-inn 8896  df-ndx 12435  df-slot 12436  df-base 12438
This theorem is referenced by:  baseslid  12488  ressvalsets  12493  ressex  12494  resseqnbasd  12501  ressressg  12503  1strbas  12542  2strbasg  12544  2stropg  12545  2strbas1g  12547  rngbaseg  12560  srngbased  12567  lmodbased  12581  ipsbased  12589  tsetndxnbasendx  12600  topgrpbasd  12606  dsndxnbasendx  12617  ring1  13049
  Copyright terms: Public domain W3C validator