Theorem ne0d 3458

Description: Deduction form of ne0i 3457. 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 3457	. 2 |- ( B e. A -> A =/= (/) )
3	1, 2	syl 14	1 |- ( ph -> A =/= (/) )

Syntax hints:   -> wi 4   e. wcel 2167   =/= wne 2367   (/)c0 3450

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-v 2765  df-dif 3159  df-nul 3451

This theorem is referenced by:  fihashelne0d  10874  mndbn0  13048  grpbn0  13138  bln0  14630