Metamath Proof Explorer


Theorem ordtr3

Description: Transitive law for ordinal classes. (Contributed by Mario Carneiro, 30-Dec-2014) (Proof shortened by JJ, 24-Sep-2021)

Ref Expression
Assertion ordtr3 Ord B Ord C A B A C C B

Proof

Step Hyp Ref Expression
1 nelss A B ¬ A C ¬ B C
2 1 adantl Ord B Ord C A B ¬ A C ¬ B C
3 ordtri1 Ord B Ord C B C ¬ C B
4 3 con2bid Ord B Ord C C B ¬ B C
5 4 adantr Ord B Ord C A B ¬ A C C B ¬ B C
6 2 5 mpbird Ord B Ord C A B ¬ A C C B
7 6 expr Ord B Ord C A B ¬ A C C B
8 7 orrd Ord B Ord C A B A C C B
9 8 ex Ord B Ord C A B A C C B