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Metamath Proof Explorer

Theorem rabsstp

Description: Conditions for a restricted class abstraction to be a subset of an unordered triple. (Contributed by Thierry Arnoux, 6-Jul-2025)

Ref Expression
Assertion rabsstp 
|- ( { x e. V | ph } C_ { X , Y , Z } <-> A. x e. V ( ph -> ( x = X \/ x = Y \/ x = Z ) ) )

Proof

Step Hyp Ref Expression
1 df-rab 
 |-  { x e. V | ph } = { x | ( x e. V /\ ph ) }
2 dftp2 
 |-  { X , Y , Z } = { x | ( x = X \/ x = Y \/ x = Z ) }
3 1 2 sseq12i 
 |-  ( { x e. V | ph } C_ { X , Y , Z } <-> { x | ( x e. V /\ ph ) } C_ { x | ( x = X \/ x = Y \/ x = Z ) } )
4 ss2ab 
 |-  ( { x | ( x e. V /\ ph ) } C_ { x | ( x = X \/ x = Y \/ x = Z ) } <-> A. x ( ( x e. V /\ ph ) -> ( x = X \/ x = Y \/ x = Z ) ) )
5 impexp 
 |-  ( ( ( x e. V /\ ph ) -> ( x = X \/ x = Y \/ x = Z ) ) <-> ( x e. V -> ( ph -> ( x = X \/ x = Y \/ x = Z ) ) ) )
6 5 albii 
 |-  ( A. x ( ( x e. V /\ ph ) -> ( x = X \/ x = Y \/ x = Z ) ) <-> A. x ( x e. V -> ( ph -> ( x = X \/ x = Y \/ x = Z ) ) ) )
7 df-ral 
 |-  ( A. x e. V ( ph -> ( x = X \/ x = Y \/ x = Z ) ) <-> A. x ( x e. V -> ( ph -> ( x = X \/ x = Y \/ x = Z ) ) ) )
8 6 7 bitr4i 
 |-  ( A. x ( ( x e. V /\ ph ) -> ( x = X \/ x = Y \/ x = Z ) ) <-> A. x e. V ( ph -> ( x = X \/ x = Y \/ x = Z ) ) )
9 3 4 8 3bitri 
 |-  ( { x e. V | ph } C_ { X , Y , Z } <-> A. x e. V ( ph -> ( x = X \/ x = Y \/ x = Z ) ) )