Theorem sssneq 14312

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 529	. . . . 5 |- ( ( A C_ { B } /\ ( y e. A /\ z e. A ) ) -> y e. A )
3	1, 2	sseldd 3154	. . . 4 |- ( ( A C_ { B } /\ ( y e. A /\ z e. A ) ) -> y e. { B } )
4		elsni 3607	. . . 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 531	. . . . 5 |- ( ( A C_ { B } /\ ( y e. A /\ z e. A ) ) -> z e. A )
7	1, 6	sseldd 3154	. . . 4 |- ( ( A C_ { B } /\ ( y e. A /\ z e. A ) ) -> z e. { B } )
8		elsni 3607	. . . 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 2211	. 2 |- ( ( A C_ { B } /\ ( y e. A /\ z e. A ) ) -> y = z )
11	10	ralrimivva 2557	1 |- ( A C_ { B } -> A. y e. A A. z e. A y = z )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2146   A.wral 2453   C_ wss 3127   {csn 3589

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-v 2737  df-in 3133  df-ss 3140  df-sn 3595

This theorem is referenced by: (None)