Theorem n0f 3558

Description: A nonempty class has at least one element. Proposition 5.17(1) of [TakeutiZaring] p. 20. This version of n0 3559 requires only that x not be free in, rather than not occur in, A. (Contributed by NM, 17-Oct-2003.)

Hypothesis

Ref	Expression
n0f.1	|- F/_xA

Assertion

Ref	Expression
n0f	|- (A =/= (/) <-> E.x x e. A)

Proof of Theorem n0f

Step	Hyp	Ref	Expression
1		n0f.1	. . . . 5 |- F/_xA
2		nfcv 2489	. . . . 5 |- F/_x(/)
3	1, 2	cleqf 2513	. . . 4 |- (A = (/) <-> A.x(x e. A <-> x e. (/)))
4		noel 3554	. . . . . 6 |- -. x e. (/)
5	4	nbn 336	. . . . 5 |- (-. x e. A <-> (x e. A <-> x e. (/)))
6	5	albii 1566	. . . 4 |- (A.x -. x e. A <-> A.x(x e. A <-> x e. (/)))
7	3, 6	bitr4i 243	. . 3 |- (A = (/) <-> A.x -. x e. A)
8	7	necon3abii 2546	. 2 |- (A =/= (/) <-> -. A.x -. x e. A)
9		df-ex 1542	. 2 |- (E.x x e. A <-> -. A.x -. x e. A)
10	8, 9	bitr4i 243	1 |- (A =/= (/) <-> E.x x e. A)

Syntax hints:  -. wn 3   <-> wb 176  A.wal 1540  E.wex 1541   = wceq 1642   e. wcel 1710   F/_wnfc 2476   =/= wne 2516  (/)c0 3550

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-nul 3551

This theorem is referenced by:  n0  3559  abn0  3568