Theorem mss 3981

Description: An inhabited class (even if proper) has an inhabited subset. (Contributed by Jim Kingdon, 17-Sep-2018.)

Assertion

Ref	Expression
mss	|- ( E. y y e. A -> E. x ( x C_ A /\ E. z z e. x ) )

Distinct variable groups:   x, y   x, z   x, A, y

Allowed substitution hint:   A( z)

Proof of Theorem mss

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2604	. . . . 5 |- y e. _V
2	1	snss 3516	. . . 4 |- ( y e. A <-> { y } C_ A )
3	1	snm 3510	. . . . 5 |- E. w w e. { y }
4	1	snex 3957	. . . . . 6 |- { y } e. _V
5		sseq1 3020	. . . . . . 7 |- ( x = { y } -> ( x C_ A <-> { y } C_ A ) )
6		eleq2 2142	. . . . . . . 8 |- ( x = { y } -> ( w e. x <-> w e. { y } ) )
7	6	exbidv 1746	. . . . . . 7 |- ( x = { y } -> ( E. w w e. x <-> E. w w e. { y } ) )
8	5, 7	anbi12d 456	. . . . . 6 |- ( x = { y } -> ( ( x C_ A /\ E. w w e. x ) <-> ( { y } C_ A /\ E. w w e. { y } ) ) )
9	4, 8	spcev 2692	. . . . 5 |- ( ( { y } C_ A /\ E. w w e. { y } ) -> E. x ( x C_ A /\ E. w w e. x ) )
10	3, 9	mpan2 415	. . . 4 |- ( { y } C_ A -> E. x ( x C_ A /\ E. w w e. x ) )
11	2, 10	sylbi 119	. . 3 |- ( y e. A -> E. x ( x C_ A /\ E. w w e. x ) )
12	11	exlimiv 1529	. 2 |- ( E. y y e. A -> E. x ( x C_ A /\ E. w w e. x ) )
13		elequ1 1640	. . . . 5 |- ( z = w -> ( z e. x <-> w e. x ) )
14	13	cbvexv 1836	. . . 4 |- ( E. z z e. x <-> E. w w e. x )
15	14	anbi2i 444	. . 3 |- ( ( x C_ A /\ E. z z e. x ) <-> ( x C_ A /\ E. w w e. x ) )
16	15	exbii 1536	. 2 |- ( E. x ( x C_ A /\ E. z z e. x ) <-> E. x ( x C_ A /\ E. w w e. x ) )
17	12, 16	sylibr 132	1 |- ( E. y y e. A -> E. x ( x C_ A /\ E. z z e. x ) )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   E.wex 1421   e. wcel 1433   C_ wss 2973   {csn 3398

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404

This theorem is referenced by: (None)