Theorem ne0f 2283

Description: A nonempty class has at least one element. Proposition 5.17(1) of [TakeutiZaring] p. 20. This version of ne0 2284 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-17 969	. . . . 5 |- (y e. (/) -> A.x y e. (/))
3	1, 2	cleqf 1557	. . . 4 |- (A = (/) <-> A.x(x e. A <-> x e. (/)))
4		noel 2280	. . . . . 6 |- -. x e. (/)
5	4	nbn 721	. . . . 5 |- (-. x e. A <-> (x e. A <-> x e. (/)))
6	5	albii 997	. . . 4 |- (A.x -. x e. A <-> A.x(x e. A <-> x e. (/)))
7	3, 6	bitr4 176	. . 3 |- (A = (/) <-> A.x -. x e. A)
8	7	negbii 187	. 2 |- (-. A = (/) <-> -. A.x -. x e. A)
9		df-ne 1584	. 2 |- (A =/= (/) <-> -. A = (/))
10		df-ex 979	. 2 |- (E.x x e. A <-> -. A.x -. x e. A)
11	8, 9, 10	3bitr4 183	1 |- (A =/= (/) <-> E.x x e. A)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146  A.wal 952   = wceq 954   e. wcel 956  E.wex 978   =/= wne 1582  (/)c0 2276

This theorem is referenced by:  ne0 2284  cp 4702

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-nul 2277

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