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Theorem n0rf 3261
 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 3262 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 𝑥𝐴𝐴 ≠ ∅)

Proof of Theorem n0rf
StepHypRef Expression
1 exalim 1432 . 2 (∃𝑥 𝑥𝐴 → ¬ ∀𝑥 ¬ 𝑥𝐴)
2 n0rf.1 . . . . 5 𝑥𝐴
3 nfcv 2220 . . . . 5 𝑥
42, 3cleqf 2243 . . . 4 (𝐴 = ∅ ↔ ∀𝑥(𝑥𝐴𝑥 ∈ ∅))
5 noel 3256 . . . . . 6 ¬ 𝑥 ∈ ∅
65nbn 648 . . . . 5 𝑥𝐴 ↔ (𝑥𝐴𝑥 ∈ ∅))
76albii 1400 . . . 4 (∀𝑥 ¬ 𝑥𝐴 ↔ ∀𝑥(𝑥𝐴𝑥 ∈ ∅))
84, 7bitr4i 185 . . 3 (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥𝐴)
98necon3abii 2282 . 2 (𝐴 ≠ ∅ ↔ ¬ ∀𝑥 ¬ 𝑥𝐴)
101, 9sylibr 132 1 (∃𝑥 𝑥𝐴𝐴 ≠ ∅)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 103  ∀wal 1283   = wceq 1285  ∃wex 1422   ∈ wcel 1434  Ⅎwnfc 2207   ≠ wne 2246  ∅c0 3252 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064 This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-v 2604  df-dif 2976  df-nul 3253 This theorem is referenced by:  n0r  3262  abn0r  3271
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