Metamath Proof Explorer


Theorem rals-no-surprise

Description: Demonstrate that there is never a "surprise" when using the allsome quantifier restricted to a class, that is, it is never possible for the consequent to be both always true and always false of the members of A that satisfy the antecedent. This is the restricted counterpart of als-no-surprise , and follows from it by dfrals2 . Note that this needs no assumption that A is nonempty, because allsome requires a member of A satisfying ph , and that member would have to satisfy both ps and -. ps . The ordinary restricted "for all" requires no such member and can be vacuously true, as shown in empty-surprise2 ; that is the point of allsome. (Contributed by David A. Wheeler, 12-Jul-2026)

Ref Expression
Assertion rals-no-surprise
|- -. ( AE x e. A ( ph -> ps ) /\ AE x e. A ( ph -> -. ps ) )

Proof

Step Hyp Ref Expression
1 als-no-surprise
 |-  -. ( AE x ( ( x e. A /\ ph ) -> ps ) /\ AE x ( ( x e. A /\ ph ) -> -. ps ) )
2 dfrals2
 |-  ( AE x e. A ( ph -> ps ) <-> AE x ( ( x e. A /\ ph ) -> ps ) )
3 dfrals2
 |-  ( AE x e. A ( ph -> -. ps ) <-> AE x ( ( x e. A /\ ph ) -> -. ps ) )
4 2 3 anbi12i
 |-  ( ( AE x e. A ( ph -> ps ) /\ AE x e. A ( ph -> -. ps ) ) <-> ( AE x ( ( x e. A /\ ph ) -> ps ) /\ AE x ( ( x e. A /\ ph ) -> -. ps ) ) )
5 1 4 mtbir
 |-  -. ( AE x e. A ( ph -> ps ) /\ AE x e. A ( ph -> -. ps ) )