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Theorem ne0d 3454
Description: Deduction form of ne0i 3453. If a class has elements, then it is nonempty. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypothesis
Ref Expression
ne0d.1  |-  ( ph  ->  B  e.  A )
Assertion
Ref Expression
ne0d  |-  ( ph  ->  A  =/=  (/) )

Proof of Theorem ne0d
StepHypRef Expression
1 ne0d.1 . 2  |-  ( ph  ->  B  e.  A )
2 ne0i 3453 . 2  |-  ( B  e.  A  ->  A  =/=  (/) )
31, 2syl 14 1  |-  ( ph  ->  A  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 2164    =/= wne 2364   (/)c0 3446
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-v 2762  df-dif 3155  df-nul 3447
This theorem is referenced by:  fihashelne0d  10858  mndbn0  13002  grpbn0  13092  bln0  14563
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