ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  n0rf GIF version

Theorem n0rf 3230
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 3231 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 1391 . 2 (∃𝑥 𝑥𝐴 → ¬ ∀𝑥 ¬ 𝑥𝐴)
2 n0rf.1 . . . . 5 𝑥𝐴
3 nfcv 2178 . . . . 5 𝑥
42, 3cleqf 2201 . . . 4 (𝐴 = ∅ ↔ ∀𝑥(𝑥𝐴𝑥 ∈ ∅))
5 noel 3225 . . . . . 6 ¬ 𝑥 ∈ ∅
65nbn 615 . . . . 5 𝑥𝐴 ↔ (𝑥𝐴𝑥 ∈ ∅))
76albii 1359 . . . 4 (∀𝑥 ¬ 𝑥𝐴 ↔ ∀𝑥(𝑥𝐴𝑥 ∈ ∅))
84, 7bitr4i 176 . . 3 (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥𝐴)
98necon3abii 2238 . 2 (𝐴 ≠ ∅ ↔ ¬ ∀𝑥 ¬ 𝑥𝐴)
101, 9sylibr 137 1 (∃𝑥 𝑥𝐴𝐴 ≠ ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 98  wal 1241   = wceq 1243  wex 1381  wcel 1393  wnfc 2165  wne 2204  c0 3221
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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-v 2556  df-dif 2917  df-nul 3222
This theorem is referenced by:  n0r  3231  abn0r  3240
  Copyright terms: Public domain W3C validator