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Theorem n0rf 3375
 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 3376 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 1478 . 2
2 n0rf.1 . . . . 5
3 nfcv 2281 . . . . 5
42, 3cleqf 2305 . . . 4
5 noel 3367 . . . . . 6
65nbn 688 . . . . 5
76albii 1446 . . . 4
84, 7bitr4i 186 . . 3
98necon3abii 2344 . 2
101, 9sylibr 133 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 104  wal 1329   wceq 1331  wex 1468   wcel 1480  wnfc 2268   wne 2308  c0 3363 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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-v 2688  df-dif 3073  df-nul 3364 This theorem is referenced by:  n0r  3376  abn0r  3387
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