Theorem nnullss 1880

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(s):   x,A

Proof of Theorem nnullss

Step	Hyp	Ref	Expression
1		n0 1714	. 2 |- (-. A = (/) <-> E.y y e. A)
2		visset 1350	. . . . 5 |- y e. V
3	2	snss 1849	. . . 4 |- (y e. A <-> {y} (_ A)
4	2	snnz 1846	. . . . 5 |- -. {y} = (/)
5		snex 1859	. . . . . 6 |- {y} e. V
6		sseq1 1521	. . . . . . 7 |- (x = {y} -> (x (_ A <-> {y} (_ A))
7		cleq1 1107	. . . . . . . 8 |- (x = {y} -> (x = (/) <-> {y} = (/)))
8	7	negbid 463	. . . . . . 7 |- (x = {y} -> (-. x = (/) <-> -. {y} = (/)))
9	6, 8	anbi12d 476	. . . . . 6 |- (x = {y} -> ((x (_ A /\ -. x = (/)) <-> ({y} (_ A /\ -. {y} = (/))))
10	5, 9	cla4ev 1401	. . . . 5 |- (({y} (_ A /\ -. {y} = (/)) -> E.x(x (_ A /\ -. x = (/)))
11	4, 10	mpan2 519	. . . 4 |- ({y} (_ A -> E.x(x (_ A /\ -. x = (/)))
12	3, 11	sylbi 174	. . 3 |- (y e. A -> E.x(x (_ A /\ -. x = (/)))
13	12	19.23aiv 952	. 2 |- (E.y y e. A -> E.x(x (_ A /\ -. x = (/)))
14	1, 13	sylbi 174	1 |- (-. A = (/) -> E.x(x (_ A /\ -. x = (/)))

Syntax hints:  -. wn 1   -> wi 2   /\ wa 196  E.wex 678   = wceq 1091   e. wcel 1092   (_ wss 1487  (/)c0 1707  {csn 1808

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812

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