ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ne0d GIF version

Theorem ne0d 3374
Description: Deduction form of ne0i 3373. If a class has elements, then it is nonempty. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypothesis
Ref Expression
ne0d.1 (𝜑𝐵𝐴)
Assertion
Ref Expression
ne0d (𝜑𝐴 ≠ ∅)

Proof of Theorem ne0d
StepHypRef Expression
1 ne0d.1 . 2 (𝜑𝐵𝐴)
2 ne0i 3373 . 2 (𝐵𝐴𝐴 ≠ ∅)
31, 2syl 14 1 (𝜑𝐴 ≠ ∅)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1481  wne 2309  c0 3367
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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-v 2691  df-dif 3077  df-nul 3368
This theorem is referenced by:  bln0  12624
  Copyright terms: Public domain W3C validator