Theorem mss 4269

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 2774	. . . . 5 |- y e. _V
2	1	snss 3767	. . . 4 |- ( y e. A <-> { y } C_ A )
3	1	snm 3752	. . . . 5 |- E. w w e. { y }
4	1	snex 4228	. . . . . 6 |- { y } e. _V
5		sseq1 3215	. . . . . . 7 |- ( x = { y } -> ( x C_ A <-> { y } C_ A ) )
6		eleq2 2268	. . . . . . . 8 |- ( x = { y } -> ( w e. x <-> w e. { y } ) )
7	6	exbidv 1847	. . . . . . 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 2867	. . . . 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 1620	. 2 |- ( E. y y e. A -> E. x ( x C_ A /\ E. w w e. x ) )
13		elequ1 2179	. . . . 5 |- ( z = w -> ( z e. x <-> w e. x ) )
14	13	cbvexv 1941	. . . 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 1627	. 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 1372   E.wex 1514   e. wcel 2175   C_ wss 3165   {csn 3632

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638

This theorem is referenced by: (None)