Theorem neq0f 3926

Description: A nonempty class has at least one element. Proposition 5.17(1) of [TakeutiZaring] p. 20. This version of neq0 3930 requires only that x not be free in, rather than not occur in, A. (Contributed by BJ, 15-Jul-2021.)

Hypothesis

Ref	Expression
eq0f.1	|- F/_ x A

Assertion

Ref	Expression
neq0f	|- ( -. A = (/) <-> E. x x e. A )

Proof of Theorem neq0f

Step	Hyp	Ref	Expression
1		eq0f.1	. . . 4 |- F/_ x A
2	1	eq0f 3925	. . 3 |- ( A = (/) <-> A. x -. x e. A )
3	2	notbii 310	. 2 |- ( -. A = (/) <-> -. A. x -. x e. A )
4		df-ex 1705	. 2 |- ( E. x x e. A <-> -. A. x -. x e. A )
5	3, 4	bitr4i 267	1 |- ( -. A = (/) <-> E. x x e. A )

Syntax hints:   -. wn 3   <-> wb 196   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   F/_wnfc 2751   (/)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-v 3202  df-dif 3577  df-nul 3916

This theorem is referenced by:  n0f  3927  neq0  3930