Theorem ne0f 2883

Description: A nonempty class has at least one element. Proposition 5.17(1) of [TakeutiZaring] p. 20. This version of n0 2884 requires only that x not be free in, rather than not occur in, A.

Hypothesis

Ref	Expression
ne0f.1	|- (y e. A -> A.x y e. A)

Assertion

Ref	Expression
ne0f	|- (A =/= (/) <-> E.x x e. A)

Distinct variable groups:   x,y   y,A

Proof of Theorem ne0f

Step	Hyp	Ref	Expression
1		ne0f.1	. . . . 5 |- (y e. A -> A.x y e. A)
2		ax-17 1317	. . . . 5 |- (y e. (/) -> A.x y e. (/))
3	1, 2	cleqf 1984	. . . 4 |- (A = (/) <-> A.x(x e. A <-> x e. (/)))
4		noel 2879	. . . . . 6 |- -. x e. (/)
5	4	nbn 791	. . . . 5 |- (-. x e. A <-> (x e. A <-> x e. (/)))
6	5	albii 1346	. . . 4 |- (A.x -. x e. A <-> A.x(x e. A <-> x e. (/)))
7	3, 6	bitr4i 193	. . 3 |- (A = (/) <-> A.x -. x e. A)
8	7	notbii 204	. 2 |- (-. A = (/) <-> -. A.x -. x e. A)
9		df-ne 2019	. 2 |- (A =/= (/) <-> -. A = (/))
10		df-ex 1327	. 2 |- (E.x x e. A <-> -. A.x -. x e. A)
11	8, 9, 10	3bitr4i 200	1 |- (A =/= (/) <-> E.x x e. A)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326   =/= wne 2017  (/)c0 2875

This theorem is referenced by:  n0 2884  cp 5852  bnj894 13327

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-nul 2876

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