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Metamath Proof Explorer

Theorem prnmax

Description: A positive real has no largest member. Definition 9-3.1(iii) of Gleason p. 121. (Contributed by NM, 9-Mar-1996) (Revised by Mario Carneiro, 11-May-2013) (New usage is discouraged.)

Ref Expression
Assertion prnmax 
|- ( ( A e. P. /\ B e. A ) -> E. x e. A B

Proof

Step Hyp Ref Expression
1 eleq1 
 |-  ( y = B -> ( y e. A <-> B e. A ) )
2 1 anbi2d 
 |-  ( y = B -> ( ( A e. P. /\ y e. A ) <-> ( A e. P. /\ B e. A ) ) )
3 breq1 
 |-  ( y = B -> ( y  B
4 3 rexbidv 
 |-  ( y = B -> ( E. x e. A y  E. x e. A B
5 2 4 imbi12d 
 |-  ( y = B -> ( ( ( A e. P. /\ y e. A ) -> E. x e. A y  ( ( A e. P. /\ B e. A ) -> E. x e. A B
6 elnpi 
 |-  ( A e. P. <-> ( ( A e. _V /\ (/) C. A /\ A C. Q. ) /\ A. y e. A ( A. x ( x  x e. A ) /\ E. x e. A y
7 6 simprbi 
 |-  ( A e. P. -> A. y e. A ( A. x ( x  x e. A ) /\ E. x e. A y
8 7 r19.21bi 
 |-  ( ( A e. P. /\ y e. A ) -> ( A. x ( x  x e. A ) /\ E. x e. A y
9 8 simprd 
 |-  ( ( A e. P. /\ y e. A ) -> E. x e. A y
10 5 9 vtoclg 
 |-  ( B e. A -> ( ( A e. P. /\ B e. A ) -> E. x e. A B
11 10 anabsi7 
 |-  ( ( A e. P. /\ B e. A ) -> E. x e. A B