Theorem n0rf 3370

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 3371 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 1478	. 2 |- ( E. x x e. A -> -. A. x -. x e. A )
2		n0rf.1	. . . . 5 |- F/_ x A
3		nfcv 2279	. . . . 5 |- F/_ x (/)
4	2, 3	cleqf 2303	. . . 4 |- ( A = (/) <-> A. x ( x e. A <-> x e. (/) ) )
5		noel 3362	. . . . . 6 |- -. x e. (/)
6	5	nbn 688	. . . . 5 |- ( -. x e. A <-> ( x e. A <-> x e. (/) ) )
7	6	albii 1446	. . . 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 2342	. 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 1329   = wceq 1331   E.wex 1468   e. wcel 1480   F/_wnfc 2266   =/= wne 2306   (/)c0 3358

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-v 2683  df-dif 3068  df-nul 3359

This theorem is referenced by:  n0r  3371  abn0r  3382