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Theorem n0rf 3211
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 3212 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 xA
Assertion
Ref Expression
n0rf (x x AA ≠ ∅)

Proof of Theorem n0rf
StepHypRef Expression
1 exalim 1371 . 2 (x x A → ¬ x ¬ x A)
2 n0rf.1 . . . . 5 xA
3 nfcv 2160 . . . . 5 x
42, 3cleqf 2183 . . . 4 (A = ∅ ↔ x(x Ax ∅))
5 noel 3206 . . . . . 6 ¬ x
65nbn 602 . . . . 5 x A ↔ (x Ax ∅))
76albii 1338 . . . 4 (x ¬ x Ax(x Ax ∅))
84, 7bitr4i 176 . . 3 (A = ∅ ↔ x ¬ x A)
98necon3abii 2217 . 2 (A ≠ ∅ ↔ ¬ x ¬ x A)
101, 9sylibr 137 1 (x x AA ≠ ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 98  wal 1314  wex 1361   = wceq 1373   wcel 1375  wnfc 2147  wne 2186  c0 3202
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 532  ax-in2 533  ax-io 617  ax-5 1315  ax-7 1316  ax-gen 1317  ax-ie1 1362  ax-ie2 1363  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-fal 1232  df-nf 1329  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-v 2535  df-dif 2898  df-nul 3203
This theorem is referenced by:  n0r  3212  abn0r  3221
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