Theorem nnullss 4654

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 3746	. 2 |- ( A =/= (/) <-> E. y y e. A )
2		vex 3073	. . . . 5 |- y e. _V
3	2	snss 4099	. . . 4 |- ( y e. A <-> { y } C_ A )
4	2	snnz 4093	. . . . 5 |- { y } =/= (/)
5		snex 4633	. . . . . 6 |- { y } e. _V
6		sseq1 3477	. . . . . . 7 |- ( x = { y } -> ( x C_ A <-> { y } C_ A ) )
7		neeq1 2729	. . . . . . 7 |- ( x = { y } -> ( x =/= (/) <-> { y } =/= (/) ) )
8	6, 7	anbi12d 710	. . . . . 6 |- ( x = { y } -> ( ( x C_ A /\ x =/= (/) ) <-> ( { y } C_ A /\ { y } =/= (/) ) ) )
9	5, 8	spcev 3162	. . . . 5 |- ( ( { y } C_ A /\ { y } =/= (/) ) -> E. x ( x C_ A /\ x =/= (/) ) )
10	4, 9	mpan2 671	. . . 4 |- ( { y } C_ A -> E. x ( x C_ A /\ x =/= (/) ) )
11	3, 10	sylbi 195	. . 3 |- ( y e. A -> E. x ( x C_ A /\ x =/= (/) ) )
12	11	exlimiv 1689	. 2 |- ( E. y y e. A -> E. x ( x C_ A /\ x =/= (/) ) )
13	1, 12	sylbi 195	1 |- ( A =/= (/) -> E. x ( x C_ A /\ x =/= (/) ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1370   E.wex 1587   e. wcel 1758   =/= wne 2644   C_ wss 3428   (/)c0 3737   {csn 3977

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pr 4631

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-v 3072  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-sn 3978  df-pr 3980

This theorem is referenced by: (None)