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Metamath Proof Explorer

Theorem hash1elsn

Description: A set of size 1 with a known element is the singleton of that element. (Contributed by Rohan Ridenour, 3-Aug-2023)

Ref Expression
Hypotheses hash1elsn.1 
|- ( ph -> ( # ` A ) = 1 )
hash1elsn.2 
|- ( ph -> B e. A )
hash1elsn.3 
|- ( ph -> A e. V )
Assertion hash1elsn 
|- ( ph -> A = { B } )

Proof

Step Hyp Ref Expression
1 hash1elsn.1 
 |-  ( ph -> ( # ` A ) = 1 )
2 hash1elsn.2 
 |-  ( ph -> B e. A )
3 hash1elsn.3 
 |-  ( ph -> A e. V )
4 hashen1 
 |-  ( A e. V -> ( ( # ` A ) = 1 <-> A ~~ 1o ) )
5 3 4 syl 
 |-  ( ph -> ( ( # ` A ) = 1 <-> A ~~ 1o ) )
6 1 5 mpbid 
 |-  ( ph -> A ~~ 1o )
7 en1 
 |-  ( A ~~ 1o <-> E. x A = { x } )
8 6 7 sylib 
 |-  ( ph -> E. x A = { x } )
9 simpr 
 |-  ( ( ph /\ A = { x } ) -> A = { x } )
10 2 adantr 
 |-  ( ( ph /\ A = { x } ) -> B e. A )
11 10 9 eleqtrd 
 |-  ( ( ph /\ A = { x } ) -> B e. { x } )
12 elsni 
 |-  ( B e. { x } -> B = x )
13 11 12 syl 
 |-  ( ( ph /\ A = { x } ) -> B = x )
14 13 sneqd 
 |-  ( ( ph /\ A = { x } ) -> { B } = { x } )
15 9 14 eqtr4d 
 |-  ( ( ph /\ A = { x } ) -> A = { B } )
16 8 15 exlimddv 
 |-  ( ph -> A = { B } )