Metamath Proof Explorer


Theorem dford2

Description: Assuming ax-reg , an ordinal is a transitive class on which inclusion satisfies trichotomy. (Contributed by Scott Fenton, 27-Oct-2010)

Ref Expression
Assertion dford2 ( Ord 𝐴 ↔ ( Tr 𝐴 ∧ ∀ 𝑥𝐴𝑦𝐴 ( 𝑥𝑦𝑥 = 𝑦𝑦𝑥 ) ) )

Proof

Step Hyp Ref Expression
1 df-ord ( Ord 𝐴 ↔ ( Tr 𝐴 ∧ E We 𝐴 ) )
2 zfregfr E Fr 𝐴
3 dfwe2 ( E We 𝐴 ↔ ( E Fr 𝐴 ∧ ∀ 𝑥𝐴𝑦𝐴 ( 𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥 ) ) )
4 2 3 mpbiran ( E We 𝐴 ↔ ∀ 𝑥𝐴𝑦𝐴 ( 𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥 ) )
5 epel ( 𝑥 E 𝑦𝑥𝑦 )
6 biid ( 𝑥 = 𝑦𝑥 = 𝑦 )
7 epel ( 𝑦 E 𝑥𝑦𝑥 )
8 5 6 7 3orbi123i ( ( 𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥 ) ↔ ( 𝑥𝑦𝑥 = 𝑦𝑦𝑥 ) )
9 8 2ralbii ( ∀ 𝑥𝐴𝑦𝐴 ( 𝑥 E 𝑦𝑥 = 𝑦𝑦 E 𝑥 ) ↔ ∀ 𝑥𝐴𝑦𝐴 ( 𝑥𝑦𝑥 = 𝑦𝑦𝑥 ) )
10 4 9 bitri ( E We 𝐴 ↔ ∀ 𝑥𝐴𝑦𝐴 ( 𝑥𝑦𝑥 = 𝑦𝑦𝑥 ) )
11 10 anbi2i ( ( Tr 𝐴 ∧ E We 𝐴 ) ↔ ( Tr 𝐴 ∧ ∀ 𝑥𝐴𝑦𝐴 ( 𝑥𝑦𝑥 = 𝑦𝑦𝑥 ) ) )
12 1 11 bitri ( Ord 𝐴 ↔ ( Tr 𝐴 ∧ ∀ 𝑥𝐴𝑦𝐴 ( 𝑥𝑦𝑥 = 𝑦𝑦𝑥 ) ) )