Theorem n0rf 3473

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 3474 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 1525	. 2 |- ( E. x x e. A -> -. A. x -. x e. A )
2		n0rf.1	. . . . 5 |- F/_ x A
3		nfcv 2348	. . . . 5 |- F/_ x (/)
4	2, 3	cleqf 2373	. . . 4 |- ( A = (/) <-> A. x ( x e. A <-> x e. (/) ) )
5		noel 3464	. . . . . 6 |- -. x e. (/)
6	5	nbn 701	. . . . 5 |- ( -. x e. A <-> ( x e. A <-> x e. (/) ) )
7	6	albii 1493	. . . 4 |- ( A. x -. x e. A <-> A. x ( x e. A <-> x e. (/) ) )
8	4, 7	bitr4i 187	. . 3 |- ( A = (/) <-> A. x -. x e. A )
9	8	necon3abii 2412	. 2 |- ( A =/= (/) <-> -. A. x -. x e. A )
10	1, 9	sylibr 134	1 |- ( E. x x e. A -> A =/= (/) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105   A.wal 1371   = wceq 1373   E.wex 1515   e. wcel 2176   F/_wnfc 2335   =/= wne 2376   (/)c0 3460

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-v 2774  df-dif 3168  df-nul 3461

This theorem is referenced by:  n0r  3474  abn0r  3485