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

Step	Hyp	Ref	Expression
1		ne0d.1	. 2 |- ( ph -> B e. A )
2		ne0i 3369	. 2 |- ( B e. A -> A =/= (/) )
3	1, 2	syl 14	1 |- ( ph -> A =/= (/) )

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  12587