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Theorem n0rf 3233
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 3234 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 1391 . 2  |-  ( E. x  x  e.  A  ->  -.  A. x  -.  x  e.  A )
2 n0rf.1 . . . . 5  |-  F/_ x A
3 nfcv 2178 . . . . 5  |-  F/_ x (/)
42, 3cleqf 2201 . . . 4  |-  ( A  =  (/)  <->  A. x ( x  e.  A  <->  x  e.  (/) ) )
5 noel 3228 . . . . . 6  |-  -.  x  e.  (/)
65nbn 615 . . . . 5  |-  ( -.  x  e.  A  <->  ( x  e.  A  <->  x  e.  (/) ) )
76albii 1359 . . . 4  |-  ( A. x  -.  x  e.  A  <->  A. x ( x  e.  A  <->  x  e.  (/) ) )
84, 7bitr4i 176 . . 3  |-  ( A  =  (/)  <->  A. x  -.  x  e.  A )
98necon3abii 2241 . 2  |-  ( A  =/=  (/)  <->  -.  A. x  -.  x  e.  A
)
101, 9sylibr 137 1  |-  ( E. x  x  e.  A  ->  A  =/=  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 98   A.wal 1241    = wceq 1243   E.wex 1381    e. wcel 1393   F/_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
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