Theorem nnullss 2763

Description: A non-empty class (even if proper) has a non-empty subset.

Assertion

Ref	Expression
nnullss	|- (A =/= (/) -> E.x(x (_ A /\ x =/= (/)))

Distinct variable group:   x,A

Proof of Theorem nnullss

Step	Hyp	Ref	Expression
1		ne0 2284	. 2 |- (A =/= (/) <-> E.y y e. A)
2		visset 1809	. . . . 5 |- y e. V
3	2	snss 2457	. . . 4 |- (y e. A <-> {y} (_ A)
4	2	snnz 2454	. . . . 5 |- {y} =/= (/)
5		snex 2745	. . . . . 6 |- {y} e. V
6		sseq1 2078	. . . . . . 7 |- (x = {y} -> (x (_ A <-> {y} (_ A))
7		neeq1 1587	. . . . . . 7 |- (x = {y} -> (x =/= (/) <-> {y} =/= (/)))
8	6, 7	anbi12d 627	. . . . . 6 |- (x = {y} -> ((x (_ A /\ x =/= (/)) <-> ({y} (_ A /\ {y} =/= (/))))
9	5, 8	cla4ev 1865	. . . . 5 |- (({y} (_ A /\ {y} =/= (/)) -> E.x(x (_ A /\ x =/= (/)))
10	4, 9	mpan2 695	. . . 4 |- ({y} (_ A -> E.x(x (_ A /\ x =/= (/)))
11	3, 10	sylbi 199	. . 3 |- (y e. A -> E.x(x (_ A /\ x =/= (/)))
12	11	19.23aiv 1293	. 2 |- (E.y y e. A -> E.x(x (_ A /\ x =/= (/)))
13	1, 12	sylbi 199	1 |- (A =/= (/) -> E.x(x (_ A /\ x =/= (/)))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 954   e. wcel 956  E.wex 978   =/= wne 1582   (_ wss 2043  (/)c0 2276  {csn 2405

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-11 965  ax-12 966  ax-13 967  ax-14 968  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  ax-sep 2698  ax-pow 2737

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409

Copyright terms: Public domain