Theorem n0fOLD 3928

Description: Obsolete proof of n0f 3927 as of 15-Jul-2021. (Contributed by NM, 17-Oct-2003.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

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

Assertion

Ref	Expression
n0fOLD	|- ( A =/= (/) <-> E. x x e. A )

Proof of Theorem n0fOLD

Step	Hyp	Ref	Expression
1		eq0f.1	. . . . 5 |- F/_ x A
2		nfcv 2764	. . . . 5 |- F/_ x (/)
3	1, 2	cleqf 2790	. . . 4 |- ( A = (/) <-> A. x ( x e. A <-> x e. (/) ) )
4		noel 3919	. . . . . 6 |- -. x e. (/)
5	4	nbn 362	. . . . 5 |- ( -. x e. A <-> ( x e. A <-> x e. (/) ) )
6	5	albii 1747	. . . 4 |- ( A. x -. x e. A <-> A. x ( x e. A <-> x e. (/) ) )
7	3, 6	bitr4i 267	. . 3 |- ( A = (/) <-> A. x -. x e. A )
8	7	necon3abii 2840	. 2 |- ( A =/= (/) <-> -. A. x -. x e. A )
9		df-ex 1705	. 2 |- ( E. x x e. A <-> -. A. x -. x e. A )
10	8, 9	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   =/= 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: (None)