Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  n0rf Unicode version

Theorem n0rf 3233
 Description: An inhabited class is nonempty. Following the Definition of [Bauer], p. 483, we call a class nonempty if and inhabited if it has at least one element. In classical logic these two concepts are equivalent, for example see Proposition 5.17(1) of [TakeutiZaring] p. 20. This version of n0r 3234 requires only that not be free in, rather than not occur in, . (Contributed by Jim Kingdon, 31-Jul-2018.)
Hypothesis
Ref Expression
n0rf.1
Assertion
Ref Expression
n0rf

Proof of Theorem n0rf
StepHypRef Expression
1 exalim 1391 . 2
2 n0rf.1 . . . . 5
3 nfcv 2178 . . . . 5
42, 3cleqf 2201 . . . 4
5 noel 3228 . . . . . 6
65nbn 615 . . . . 5
76albii 1359 . . . 4
84, 7bitr4i 176 . . 3
98necon3abii 2241 . 2
101, 9sylibr 137 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 98  wal 1241   wceq 1243  wex 1381   wcel 1393  wnfc 2165   wne 2204  c0 3224 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-v 2559  df-dif 2920  df-nul 3225 This theorem is referenced by:  n0r  3234  abn0r  3243
 Copyright terms: Public domain W3C validator