Theorem ne0d 3442

Description: Deduction form of ne0i 3441. 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 3441	. 2 |- ( B e. A -> A =/= (/) )
3	1, 2	syl 14	1 |- ( ph -> A =/= (/) )

Syntax hints:   -> wi 4   e. wcel 2158   =/= wne 2357   (/)c0 3434

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-v 2751  df-dif 3143  df-nul 3435

This theorem is referenced by:  fihashelne0d  10791  mndbn0  12854  grpbn0  12927  bln0  14214