Theorem ne0d 39308

Description: If a set has elements, then it is not empty. (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 3921	. 2 |- ( B e. A -> A =/= (/) )
3	1, 2	syl 17	1 |- ( ph -> A =/= (/) )

Syntax hints:   -> wi 4   e. wcel 1990   =/= wne 2794   (/)c0 3915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-v 3202  df-dif 3577  df-nul 3916

This theorem is referenced by:  uzn0d  39652  uzublem  39657  climinf2lem  39938  cnrefiisplem  40055  smfsuplem1  41017  smfsuplem3  41019