Theorem mss 4318

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 2805	. . . . 5 |- y e. _V
2	1	snss 3808	. . . 4 |- ( y e. A <-> { y } C_ A )
3	1	snm 3792	. . . . 5 |- E. w w e. { y }
4	1	snex 4275	. . . . . 6 |- { y } e. _V
5		sseq1 3250	. . . . . . 7 |- ( x = { y } -> ( x C_ A <-> { y } C_ A ) )
6		eleq2 2295	. . . . . . . 8 |- ( x = { y } -> ( w e. x <-> w e. { y } ) )
7	6	exbidv 1873	. . . . . . 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 2901	. . . . 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 1646	. 2 |- ( E. y y e. A -> E. x ( x C_ A /\ E. w w e. x ) )
13		elequ1 2206	. . . . 5 |- ( z = w -> ( z e. x <-> w e. x ) )
14	13	cbvexv 1967	. . . 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 1653	. 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 1397   E.wex 1540   e. wcel 2202   C_ wss 3200   {csn 3669

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675

This theorem is referenced by: (None)