Theorem mss 3962

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 2560	. . . . 5 |- y e. _V
2	1	snss 3494	. . . 4 |- ( y e. A <-> { y } C_ A )
3	1	snm 3488	. . . . 5 |- E. w w e. { y }
4		snexgOLD 3935	. . . . . . 7 |- ( y e. _V -> { y } e. _V )
5	1, 4	ax-mp 7	. . . . . 6 |- { y } e. _V
6		sseq1 2966	. . . . . . 7 |- ( x = { y } -> ( x C_ A <-> { y } C_ A ) )
7		eleq2 2101	. . . . . . . 8 |- ( x = { y } -> ( w e. x <-> w e. { y } ) )
8	7	exbidv 1706	. . . . . . 7 |- ( x = { y } -> ( E. w w e. x <-> E. w w e. { y } ) )
9	6, 8	anbi12d 442	. . . . . 6 |- ( x = { y } -> ( ( x C_ A /\ E. w w e. x ) <-> ( { y } C_ A /\ E. w w e. { y } ) ) )
10	5, 9	spcev 2647	. . . . 5 |- ( ( { y } C_ A /\ E. w w e. { y } ) -> E. x ( x C_ A /\ E. w w e. x ) )
11	3, 10	mpan2 401	. . . 4 |- ( { y } C_ A -> E. x ( x C_ A /\ E. w w e. x ) )
12	2, 11	sylbi 114	. . 3 |- ( y e. A -> E. x ( x C_ A /\ E. w w e. x ) )
13	12	exlimiv 1489	. 2 |- ( E. y y e. A -> E. x ( x C_ A /\ E. w w e. x ) )
14		elequ1 1600	. . . . 5 |- ( z = w -> ( z e. x <-> w e. x ) )
15	14	cbvexv 1795	. . . 4 |- ( E. z z e. x <-> E. w w e. x )
16	15	anbi2i 430	. . 3 |- ( ( x C_ A /\ E. z z e. x ) <-> ( x C_ A /\ E. w w e. x ) )
17	16	exbii 1496	. 2 |- ( E. x ( x C_ A /\ E. z z e. x ) <-> E. x ( x C_ A /\ E. w w e. x ) )
18	13, 17	sylibr 137	1 |- ( E. y y e. A -> E. x ( x C_ A /\ E. z z e. x ) )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243   E.wex 1381   e. wcel 1393   _Vcvv 2557   C_ wss 2917   {csn 3375

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381

This theorem is referenced by: (None)