Theorem n0rf 3421

Description: An inhabited class is nonempty. Following the Definition of [Bauer], p. 483, we call a class A nonempty if A =/= (/) and inhabited if it has at least one element. In classical logic these two concepts are equivalent, for example see Proposition 5.17(1) of [TakeutiZaring] p. 20. This version of n0r 3422 requires only that x not be free in, rather than not occur in, A. (Contributed by Jim Kingdon, 31-Jul-2018.)

Hypothesis

Ref	Expression
n0rf.1	|- F/_ x A

Assertion

Ref	Expression
n0rf	|- ( E. x x e. A -> A =/= (/) )

Proof of Theorem n0rf

Step	Hyp	Ref	Expression
1		exalim 1490	. 2 |- ( E. x x e. A -> -. A. x -. x e. A )
2		n0rf.1	. . . . 5 |- F/_ x A
3		nfcv 2308	. . . . 5 |- F/_ x (/)
4	2, 3	cleqf 2333	. . . 4 |- ( A = (/) <-> A. x ( x e. A <-> x e. (/) ) )
5		noel 3413	. . . . . 6 |- -. x e. (/)
6	5	nbn 689	. . . . 5 |- ( -. x e. A <-> ( x e. A <-> x e. (/) ) )
7	6	albii 1458	. . . 4 |- ( A. x -. x e. A <-> A. x ( x e. A <-> x e. (/) ) )
8	4, 7	bitr4i 186	. . 3 |- ( A = (/) <-> A. x -. x e. A )
9	8	necon3abii 2372	. 2 |- ( A =/= (/) <-> -. A. x -. x e. A )
10	1, 9	sylibr 133	1 |- ( E. x x e. A -> A =/= (/) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 104   A.wal 1341   = wceq 1343   E.wex 1480   e. wcel 2136   F/_wnfc 2295   =/= wne 2336   (/)c0 3409

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-v 2728  df-dif 3118  df-nul 3410

This theorem is referenced by:  n0r  3422  abn0r  3433