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Theorem n0rf 3227
 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 3228 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 1388 . 2 (x x A → ¬ x ¬ x A)
2 n0rf.1 . . . . 5 xA
3 nfcv 2175 . . . . 5 x
42, 3cleqf 2198 . . . 4 (A = ∅ ↔ x(x Ax ∅))
5 noel 3222 . . . . . 6 ¬ x
65nbn 614 . . . . 5 x A ↔ (x Ax ∅))
76albii 1356 . . . 4 (x ¬ x Ax(x Ax ∅))
84, 7bitr4i 176 . . 3 (A = ∅ ↔ x ¬ x A)
98necon3abii 2235 . 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 1240   = wceq 1242  ∃wex 1378   ∈ wcel 1390  Ⅎwnfc 2162   ≠ wne 2201  ∅c0 3218 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-v 2553  df-dif 2914  df-nul 3219 This theorem is referenced by:  n0r  3228  abn0r  3237
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