Theorem sssneq 16603

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 109	. . . . 5 |- ( ( A C_ { B } /\ ( y e. A /\ z e. A ) ) -> A C_ { B } )
2		simprl 531	. . . . 5 |- ( ( A C_ { B } /\ ( y e. A /\ z e. A ) ) -> y e. A )
3	1, 2	sseldd 3228	. . . 4 |- ( ( A C_ { B } /\ ( y e. A /\ z e. A ) ) -> y e. { B } )
4		elsni 3687	. . . 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 533	. . . . 5 |- ( ( A C_ { B } /\ ( y e. A /\ z e. A ) ) -> z e. A )
7	1, 6	sseldd 3228	. . . 4 |- ( ( A C_ { B } /\ ( y e. A /\ z e. A ) ) -> z e. { B } )
8		elsni 3687	. . . 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 2267	. 2 |- ( ( A C_ { B } /\ ( y e. A /\ z e. A ) ) -> y = z )
11	10	ralrimivva 2614	1 |- ( A C_ { B } -> A. y e. A A. z e. A y = z )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   A.wral 2510   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-ext 2213

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-ral 2515  df-v 2804  df-in 3206  df-ss 3213  df-sn 3675

This theorem is referenced by: (None)