Theorem mss 4361

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 2824	. . . . 5 |- y e. _V
2	1	snss 3845	. . . 4 |- ( y e. A <-> { y } C_ A )
3	1	snm 3828	. . . . 5 |- E. w w e. { y }
4	1	snex 4317	. . . . . 6 |- { y } e. _V
5		sseq1 3271	. . . . . . 7 |- ( x = { y } -> ( x C_ A <-> { y } C_ A ) )
6		eleq2 2302	. . . . . . . 8 |- ( x = { y } -> ( w e. x <-> w e. { y } ) )
7	6	exbidv 1878	. . . . . . 7 |- ( x = { y } -> ( E. w w e. x <-> E. w w e. { y } ) )
8	5, 7	anbi12d 477	. . . . . 6 |- ( x = { y } -> ( ( x C_ A /\ E. w w e. x ) <-> ( { y } C_ A /\ E. w w e. { y } ) ) )
9	4, 8	spcev 2920	. . . . 5 |- ( ( { y } C_ A /\ E. w w e. { y } ) -> E. x ( x C_ A /\ E. w w e. x ) )
10	3, 9	mpan2 429	. . . 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 1651	. 2 |- ( E. y y e. A -> E. x ( x C_ A /\ E. w w e. x ) )
13		elequ1 2213	. . . . 5 |- ( z = w -> ( z e. x <-> w e. x ) )
14	13	cbvexv 1974	. . . 4 |- ( E. z z e. x <-> E. w w e. x )
15	14	anbi2i 461	. . 3 |- ( ( x C_ A /\ E. z z e. x ) <-> ( x C_ A /\ E. w w e. x ) )
16	15	exbii 1658	. 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 1402   E.wex 1545   e. wcel 2209   C_ wss 3220   {csn 3705

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-13 2211  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711

This theorem is referenced by: (None)