Metamath Proof Explorer

Theorem 7olt9o

Description: Ordinal 7 is less than ordinal 9. (Contributed by BTernaryTau, 17-Oct-2026)

Ref Expression
Assertion 7olt9o 
|- 7o e. 9o

Proof

Step Hyp Ref Expression
1 7on 
 |-  7o e. On
2 sucidg 
 |-  ( 7o e. On -> 7o e. suc 7o )
3 1 2 ax-mp 
 |-  7o e. suc 7o
4 df-8o 
 |-  8o = suc 7o
5 3 4 eleqtrri 
 |-  7o e. 8o
6 8olt9o 
 |-  8o e. 9o
7 9on 
 |-  9o e. On
8 ontr1 
 |-  ( 9o e. On -> ( ( 7o e. 8o /\ 8o e. 9o ) -> 7o e. 9o ) )
9 7 8 ax-mp 
 |-  ( ( 7o e. 8o /\ 8o e. 9o ) -> 7o e. 9o )
10 5 6 9 mp2an 
 |-  7o e. 9o