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Theorem n0rf 3314
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 3315 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 1443 . 2 (∃𝑥 𝑥𝐴 → ¬ ∀𝑥 ¬ 𝑥𝐴)
2 n0rf.1 . . . . 5 𝑥𝐴
3 nfcv 2235 . . . . 5 𝑥
42, 3cleqf 2259 . . . 4 (𝐴 = ∅ ↔ ∀𝑥(𝑥𝐴𝑥 ∈ ∅))
5 noel 3306 . . . . . 6 ¬ 𝑥 ∈ ∅
65nbn 653 . . . . 5 𝑥𝐴 ↔ (𝑥𝐴𝑥 ∈ ∅))
76albii 1411 . . . 4 (∀𝑥 ¬ 𝑥𝐴 ↔ ∀𝑥(𝑥𝐴𝑥 ∈ ∅))
84, 7bitr4i 186 . . 3 (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥𝐴)
98necon3abii 2298 . 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 1294   = wceq 1296  wex 1433  wcel 1445  wnfc 2222  wne 2262  c0 3302
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-v 2635  df-dif 3015  df-nul 3303
This theorem is referenced by:  n0r  3315  abn0r  3326
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