Theorem nnullss 4930

Description: A nonempty class (even if proper) has a nonempty subset. (Contributed by NM, 23-Aug-2003.)

Assertion

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

Distinct variable group:   x, A

Proof of Theorem nnullss

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		n0 3931	. 2 |- ( A =/= (/) <-> E. y y e. A )
2		vex 3203	. . . . 5 |- y e. _V
3	2	snss 4316	. . . 4 |- ( y e. A <-> { y } C_ A )
4	2	snnz 4309	. . . . 5 |- { y } =/= (/)
5		snex 4908	. . . . . 6 |- { y } e. _V
6		sseq1 3626	. . . . . . 7 |- ( x = { y } -> ( x C_ A <-> { y } C_ A ) )
7		neeq1 2856	. . . . . . 7 |- ( x = { y } -> ( x =/= (/) <-> { y } =/= (/) ) )
8	6, 7	anbi12d 747	. . . . . 6 |- ( x = { y } -> ( ( x C_ A /\ x =/= (/) ) <-> ( { y } C_ A /\ { y } =/= (/) ) ) )
9	5, 8	spcev 3300	. . . . 5 |- ( ( { y } C_ A /\ { y } =/= (/) ) -> E. x ( x C_ A /\ x =/= (/) ) )
10	4, 9	mpan2 707	. . . 4 |- ( { y } C_ A -> E. x ( x C_ A /\ x =/= (/) ) )
11	3, 10	sylbi 207	. . 3 |- ( y e. A -> E. x ( x C_ A /\ x =/= (/) ) )
12	11	exlimiv 1858	. 2 |- ( E. y y e. A -> E. x ( x C_ A /\ x =/= (/) ) )
13	1, 12	sylbi 207	1 |- ( A =/= (/) -> E. x ( x C_ A /\ x =/= (/) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   C_ wss 3574   (/)c0 3915   {csn 4177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178  df-pr 4180

This theorem is referenced by: (None)