Theorem sssneq 13882

Description: Any two elements of a subset of a singleton are equal. (Contributed by Jim Kingdon, 28-May-2024.)

Assertion

Ref	Expression
sssneq	|- ( A C_ { B } -> A. y e. A A. z e. A y = z )

Distinct variable groups:   y, A, z   y, B, z

Proof of Theorem sssneq

Step	Hyp	Ref	Expression
1		simpl 108	. . . . 5 |- ( ( A C_ { B } /\ ( y e. A /\ z e. A ) ) -> A C_ { B } )
2		simprl 521	. . . . 5 |- ( ( A C_ { B } /\ ( y e. A /\ z e. A ) ) -> y e. A )
3	1, 2	sseldd 3143	. . . 4 |- ( ( A C_ { B } /\ ( y e. A /\ z e. A ) ) -> y e. { B } )
4		elsni 3594	. . . 4 |- ( y e. { B } -> y = B )
5	3, 4	syl 14	. . 3 |- ( ( A C_ { B } /\ ( y e. A /\ z e. A ) ) -> y = B )
6		simprr 522	. . . . 5 |- ( ( A C_ { B } /\ ( y e. A /\ z e. A ) ) -> z e. A )
7	1, 6	sseldd 3143	. . . 4 |- ( ( A C_ { B } /\ ( y e. A /\ z e. A ) ) -> z e. { B } )
8		elsni 3594	. . . 4 |- ( z e. { B } -> z = B )
9	7, 8	syl 14	. . 3 |- ( ( A C_ { B } /\ ( y e. A /\ z e. A ) ) -> z = B )
10	5, 9	eqtr4d 2201	. 2 |- ( ( A C_ { B } /\ ( y e. A /\ z e. A ) ) -> y = z )
11	10	ralrimivva 2548	1 |- ( A C_ { B } -> A. y e. A A. z e. A y = z )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   A.wral 2444   C_ wss 3116   {csn 3576

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-v 2728  df-in 3122  df-ss 3129  df-sn 3582

This theorem is referenced by: (None)