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Theorem ne0d 3370
Description: Deduction form of ne0i 3369. 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 3369 . 2  |-  ( B  e.  A  ->  A  =/=  (/) )
31, 2syl 14 1  |-  ( ph  ->  A  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1480    =/= wne 2308   (/)c0 3363
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-v 2688  df-dif 3073  df-nul 3364
This theorem is referenced by:  bln0  12601
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