Metamath Proof Explorer


Theorem gtnelioc

Description: A real number larger than the upper bound of a left-open right-closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 11-Dec-2019)

Ref Expression
Hypotheses gtnelioc.a φ A *
gtnelioc.b φ B
gtnelioc.c φ C *
gtnelioc.bltc φ B < C
Assertion gtnelioc φ ¬ C A B

Proof

Step Hyp Ref Expression
1 gtnelioc.a φ A *
2 gtnelioc.b φ B
3 gtnelioc.c φ C *
4 gtnelioc.bltc φ B < C
5 2 rexrd φ B *
6 xrltnle B * C * B < C ¬ C B
7 5 3 6 syl2anc φ B < C ¬ C B
8 4 7 mpbid φ ¬ C B
9 8 intn3an3d φ ¬ C A < C C B
10 elioc2 A * B C A B C A < C C B
11 1 2 10 syl2anc φ C A B C A < C C B
12 9 11 mtbird φ ¬ C A B