Metamath Proof Explorer

Theorem el7g

Description: Members of a set with seven elements. Lemma for usgrexmpl2nb0 etc. (Contributed by AV, 9-Aug-2025)

Ref Expression
Assertion el7g 
|- ( X e. V -> ( X e. ( { A } u. ( { B , C , D } u. { E , F , G } ) ) <-> ( X = A \/ ( ( X = B \/ X = C \/ X = D ) \/ ( X = E \/ X = F \/ X = G ) ) ) ) )

Proof

Step Hyp Ref Expression
1 elun 
 |-  ( X e. ( { A } u. ( { B , C , D } u. { E , F , G } ) ) <-> ( X e. { A } \/ X e. ( { B , C , D } u. { E , F , G } ) ) )
2 elsng 
 |-  ( X e. V -> ( X e. { A } <-> X = A ) )
3 elun 
 |-  ( X e. ( { B , C , D } u. { E , F , G } ) <-> ( X e. { B , C , D } \/ X e. { E , F , G } ) )
4 eltpg 
 |-  ( X e. V -> ( X e. { B , C , D } <-> ( X = B \/ X = C \/ X = D ) ) )
5 eltpg 
 |-  ( X e. V -> ( X e. { E , F , G } <-> ( X = E \/ X = F \/ X = G ) ) )
6 4 5 orbi12d 
 |-  ( X e. V -> ( ( X e. { B , C , D } \/ X e. { E , F , G } ) <-> ( ( X = B \/ X = C \/ X = D ) \/ ( X = E \/ X = F \/ X = G ) ) ) )
7 3 6 bitrid 
 |-  ( X e. V -> ( X e. ( { B , C , D } u. { E , F , G } ) <-> ( ( X = B \/ X = C \/ X = D ) \/ ( X = E \/ X = F \/ X = G ) ) ) )
8 2 7 orbi12d 
 |-  ( X e. V -> ( ( X e. { A } \/ X e. ( { B , C , D } u. { E , F , G } ) ) <-> ( X = A \/ ( ( X = B \/ X = C \/ X = D ) \/ ( X = E \/ X = F \/ X = G ) ) ) ) )
9 1 8 bitrid 
 |-  ( X e. V -> ( X e. ( { A } u. ( { B , C , D } u. { E , F , G } ) ) <-> ( X = A \/ ( ( X = B \/ X = C \/ X = D ) \/ ( X = E \/ X = F \/ X = G ) ) ) ) )