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Theorem n0rf 3345
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 3346 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 1463 . 2 (∃𝑥 𝑥𝐴 → ¬ ∀𝑥 ¬ 𝑥𝐴)
2 n0rf.1 . . . . 5 𝑥𝐴
3 nfcv 2258 . . . . 5 𝑥
42, 3cleqf 2282 . . . 4 (𝐴 = ∅ ↔ ∀𝑥(𝑥𝐴𝑥 ∈ ∅))
5 noel 3337 . . . . . 6 ¬ 𝑥 ∈ ∅
65nbn 673 . . . . 5 𝑥𝐴 ↔ (𝑥𝐴𝑥 ∈ ∅))
76albii 1431 . . . 4 (∀𝑥 ¬ 𝑥𝐴 ↔ ∀𝑥(𝑥𝐴𝑥 ∈ ∅))
84, 7bitr4i 186 . . 3 (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥𝐴)
98necon3abii 2321 . 2 (𝐴 ≠ ∅ ↔ ¬ ∀𝑥 ¬ 𝑥𝐴)
101, 9sylibr 133 1 (∃𝑥 𝑥𝐴𝐴 ≠ ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 104  wal 1314   = wceq 1316  wex 1453  wcel 1465  wnfc 2245  wne 2285  c0 3333
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-v 2662  df-dif 3043  df-nul 3334
This theorem is referenced by:  n0r  3346  abn0r  3357
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