Theorem n0rf 3260

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 3261 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 1431	. 2 |- ( E. x x e. A -> -. A. x -. x e. A )
2		n0rf.1	. . . . 5 |- F/_ x A
3		nfcv 2219	. . . . 5 |- F/_ x (/)
4	2, 3	cleqf 2242	. . . 4 |- ( A = (/) <-> A. x ( x e. A <-> x e. (/) ) )
5		noel 3255	. . . . . 6 |- -. x e. (/)
6	5	nbn 647	. . . . 5 |- ( -. x e. A <-> ( x e. A <-> x e. (/) ) )
7	6	albii 1399	. . . 4 |- ( A. x -. x e. A <-> A. x ( x e. A <-> x e. (/) ) )
8	4, 7	bitr4i 185	. . 3 |- ( A = (/) <-> A. x -. x e. A )
9	8	necon3abii 2281	. 2 |- ( A =/= (/) <-> -. A. x -. x e. A )
10	1, 9	sylibr 132	1 |- ( E. x x e. A -> A =/= (/) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 103   A.wal 1282   = wceq 1284   E.wex 1421   e. wcel 1433   F/_wnfc 2206   =/= wne 2245   (/)c0 3251

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-v 2603  df-dif 2975  df-nul 3252

This theorem is referenced by:  n0r  3261  abn0r  3270