Theorem nnullss 4699

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 3793	. 2 |- ( A =/= (/) <-> E. y y e. A )
2		vex 3109	. . . . 5 |- y e. _V
3	2	snss 4140	. . . 4 |- ( y e. A <-> { y } C_ A )
4	2	snnz 4134	. . . . 5 |- { y } =/= (/)
5		snex 4678	. . . . . 6 |- { y } e. _V
6		sseq1 3510	. . . . . . 7 |- ( x = { y } -> ( x C_ A <-> { y } C_ A ) )
7		neeq1 2735	. . . . . . 7 |- ( x = { y } -> ( x =/= (/) <-> { y } =/= (/) ) )
8	6, 7	anbi12d 708	. . . . . 6 |- ( x = { y } -> ( ( x C_ A /\ x =/= (/) ) <-> ( { y } C_ A /\ { y } =/= (/) ) ) )
9	5, 8	spcev 3198	. . . . 5 |- ( ( { y } C_ A /\ { y } =/= (/) ) -> E. x ( x C_ A /\ x =/= (/) ) )
10	4, 9	mpan2 669	. . . 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 1727	. 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 367   = wceq 1398   E.wex 1617   e. wcel 1823   =/= wne 2649   C_ wss 3461   (/)c0 3783   {csn 4016

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-sn 4017  df-pr 4019

This theorem is referenced by: (None)