Theorem n0rf 3314

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 3315 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 1443	. 2 |- ( E. x x e. A -> -. A. x -. x e. A )
2		n0rf.1	. . . . 5 |- F/_ x A
3		nfcv 2235	. . . . 5 |- F/_ x (/)
4	2, 3	cleqf 2259	. . . 4 |- ( A = (/) <-> A. x ( x e. A <-> x e. (/) ) )
5		noel 3306	. . . . . 6 |- -. x e. (/)
6	5	nbn 653	. . . . 5 |- ( -. x e. A <-> ( x e. A <-> x e. (/) ) )
7	6	albii 1411	. . . 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 2298	. 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 1294   = wceq 1296   E.wex 1433   e. wcel 1445   F/_wnfc 2222   =/= wne 2262   (/)c0 3302

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ne 2263  df-v 2635  df-dif 3015  df-nul 3303

This theorem is referenced by:  n0r  3315  abn0r  3326