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Theorem n0rf 3447
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 3448 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 1512 . 2 (∃𝑥 𝑥𝐴 → ¬ ∀𝑥 ¬ 𝑥𝐴)
2 n0rf.1 . . . . 5 𝑥𝐴
3 nfcv 2329 . . . . 5 𝑥
42, 3cleqf 2354 . . . 4 (𝐴 = ∅ ↔ ∀𝑥(𝑥𝐴𝑥 ∈ ∅))
5 noel 3438 . . . . . 6 ¬ 𝑥 ∈ ∅
65nbn 700 . . . . 5 𝑥𝐴 ↔ (𝑥𝐴𝑥 ∈ ∅))
76albii 1480 . . . 4 (∀𝑥 ¬ 𝑥𝐴 ↔ ∀𝑥(𝑥𝐴𝑥 ∈ ∅))
84, 7bitr4i 187 . . 3 (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥𝐴)
98necon3abii 2393 . 2 (𝐴 ≠ ∅ ↔ ¬ ∀𝑥 ¬ 𝑥𝐴)
101, 9sylibr 134 1 (∃𝑥 𝑥𝐴𝐴 ≠ ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 105  wal 1361   = wceq 1363  wex 1502  wcel 2158  wnfc 2316  wne 2357  c0 3434
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-v 2751  df-dif 3143  df-nul 3435
This theorem is referenced by:  n0r  3448  abn0r  3459
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