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Theorem n0rf 3375
Description: An inhabited class is nonempty. Following the Definition of [Bauer], p. 483, we call a class  A nonempty if  A  =/=  (/) 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  x not be free in, rather than not occur in,  A. (Contributed by Jim Kingdon, 31-Jul-2018.)
Hypothesis
Ref Expression
n0rf.1  |-  F/_ x A
Assertion
Ref Expression
n0rf  |-  ( E. x  x  e.  A  ->  A  =/=  (/) )

Proof of Theorem n0rf
StepHypRef Expression
1 exalim 1478 . 2  |-  ( E. x  x  e.  A  ->  -.  A. x  -.  x  e.  A )
2 n0rf.1 . . . . 5  |-  F/_ x A
3 nfcv 2281 . . . . 5  |-  F/_ x (/)
42, 3cleqf 2305 . . . 4  |-  ( A  =  (/)  <->  A. x ( x  e.  A  <->  x  e.  (/) ) )
5 noel 3367 . . . . . 6  |-  -.  x  e.  (/)
65nbn 688 . . . . 5  |-  ( -.  x  e.  A  <->  ( x  e.  A  <->  x  e.  (/) ) )
76albii 1446 . . . 4  |-  ( A. x  -.  x  e.  A  <->  A. x ( x  e.  A  <->  x  e.  (/) ) )
84, 7bitr4i 186 . . 3  |-  ( A  =  (/)  <->  A. x  -.  x  e.  A )
98necon3abii 2344 . 2  |-  ( A  =/=  (/)  <->  -.  A. x  -.  x  e.  A
)
101, 9sylibr 133 1  |-  ( E. x  x  e.  A  ->  A  =/=  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 104   A.wal 1329    = wceq 1331   E.wex 1468    e. wcel 1480   F/_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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