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Metamath Proof Explorer

Theorem rmoun

Description: "At most one" restricted existential quantifier for a union implies the same quantifier on both sets. (Contributed by Thierry Arnoux, 27-Nov-2023)

Ref Expression
Assertion rmoun 
|- ( E* x e. ( A u. B ) ph -> ( E* x e. A ph /\ E* x e. B ph ) )

Proof

Step Hyp Ref Expression
1 mooran2 
 |-  ( E* x ( ( x e. A /\ ph ) \/ ( x e. B /\ ph ) ) -> ( E* x ( x e. A /\ ph ) /\ E* x ( x e. B /\ ph ) ) )
2 df-rmo 
 |-  ( E* x e. ( A u. B ) ph <-> E* x ( x e. ( A u. B ) /\ ph ) )
3 elun 
 |-  ( x e. ( A u. B ) <-> ( x e. A \/ x e. B ) )
4 3 anbi1i 
 |-  ( ( x e. ( A u. B ) /\ ph ) <-> ( ( x e. A \/ x e. B ) /\ ph ) )
5 andir 
 |-  ( ( ( x e. A \/ x e. B ) /\ ph ) <-> ( ( x e. A /\ ph ) \/ ( x e. B /\ ph ) ) )
6 4 5 bitri 
 |-  ( ( x e. ( A u. B ) /\ ph ) <-> ( ( x e. A /\ ph ) \/ ( x e. B /\ ph ) ) )
7 6 mobii 
 |-  ( E* x ( x e. ( A u. B ) /\ ph ) <-> E* x ( ( x e. A /\ ph ) \/ ( x e. B /\ ph ) ) )
8 2 7 bitri 
 |-  ( E* x e. ( A u. B ) ph <-> E* x ( ( x e. A /\ ph ) \/ ( x e. B /\ ph ) ) )
9 df-rmo 
 |-  ( E* x e. A ph <-> E* x ( x e. A /\ ph ) )
10 df-rmo 
 |-  ( E* x e. B ph <-> E* x ( x e. B /\ ph ) )
11 9 10 anbi12i 
 |-  ( ( E* x e. A ph /\ E* x e. B ph ) <-> ( E* x ( x e. A /\ ph ) /\ E* x ( x e. B /\ ph ) ) )
12 1 8 11 3imtr4i 
 |-  ( E* x e. ( A u. B ) ph -> ( E* x e. A ph /\ E* x e. B ph ) )