Theorem nnullss 2768

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 2288	. 2 |- (A =/= (/) <-> E.y y e. A)
2		visset 1813	. . . . 5 |- y e. V
3	2	snss 2461	. . . 4 |- (y e. A <-> {y} (_ A)
4	2	snnz 2458	. . . . 5 |- {y} =/= (/)
5		snex 2750	. . . . . 6 |- {y} e. V
6		sseq1 2082	. . . . . . 7 |- (x = {y} -> (x (_ A <-> {y} (_ A))
7		neeq1 1590	. . . . . . 7 |- (x = {y} -> (x =/= (/) <-> {y} =/= (/)))
8	6, 7	anbi12d 628	. . . . . 6 |- (x = {y} -> ((x (_ A /\ x =/= (/)) <-> ({y} (_ A /\ {y} =/= (/))))
9	5, 8	cla4ev 1869	. . . . 5 |- (({y} (_ A /\ {y} =/= (/)) -> E.x(x (_ A /\ x =/= (/)))
10	4, 9	mpan2 696	. . . 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 1295	. 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 956   e. wcel 958  E.wex 980   =/= wne 1585   (_ wss 2047  (/)c0 2280  {csn 2409

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413

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