Theorem mss 4260

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 2766	. . . . 5 |- y e. _V
2	1	snss 3758	. . . 4 |- ( y e. A <-> { y } C_ A )
3	1	snm 3743	. . . . 5 |- E. w w e. { y }
4	1	snex 4219	. . . . . 6 |- { y } e. _V
5		sseq1 3207	. . . . . . 7 |- ( x = { y } -> ( x C_ A <-> { y } C_ A ) )
6		eleq2 2260	. . . . . . . 8 |- ( x = { y } -> ( w e. x <-> w e. { y } ) )
7	6	exbidv 1839	. . . . . . 7 |- ( x = { y } -> ( E. w w e. x <-> E. w w e. { y } ) )
8	5, 7	anbi12d 473	. . . . . 6 |- ( x = { y } -> ( ( x C_ A /\ E. w w e. x ) <-> ( { y } C_ A /\ E. w w e. { y } ) ) )
9	4, 8	spcev 2859	. . . . 5 |- ( ( { y } C_ A /\ E. w w e. { y } ) -> E. x ( x C_ A /\ E. w w e. x ) )
10	3, 9	mpan2 425	. . . 4 |- ( { y } C_ A -> E. x ( x C_ A /\ E. w w e. x ) )
11	2, 10	sylbi 121	. . 3 |- ( y e. A -> E. x ( x C_ A /\ E. w w e. x ) )
12	11	exlimiv 1612	. 2 |- ( E. y y e. A -> E. x ( x C_ A /\ E. w w e. x ) )
13		elequ1 2171	. . . . 5 |- ( z = w -> ( z e. x <-> w e. x ) )
14	13	cbvexv 1933	. . . 4 |- ( E. z z e. x <-> E. w w e. x )
15	14	anbi2i 457	. . 3 |- ( ( x C_ A /\ E. z z e. x ) <-> ( x C_ A /\ E. w w e. x ) )
16	15	exbii 1619	. 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 134	1 |- ( E. y y e. A -> E. x ( x C_ A /\ E. z z e. x ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   E.wex 1506   e. wcel 2167   C_ wss 3157   {csn 3623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629

This theorem is referenced by: (None)