Metamath Proof Explorer


Theorem rankr1bg

Description: A relationship between rank and R1 . See rankr1ag for the membership version. (Contributed by Mario Carneiro, 17-Nov-2014)

Ref Expression
Assertion rankr1bg AR1OnBdomR1AR1BrankAB

Proof

Step Hyp Ref Expression
1 r1funlim FunR1LimdomR1
2 1 simpri LimdomR1
3 limsuc LimdomR1BdomR1sucBdomR1
4 2 3 ax-mp BdomR1sucBdomR1
5 rankr1ag AR1OnsucBdomR1AR1sucBrankAsucB
6 4 5 sylan2b AR1OnBdomR1AR1sucBrankAsucB
7 r1sucg BdomR1R1sucB=𝒫R1B
8 7 adantl AR1OnBdomR1R1sucB=𝒫R1B
9 8 eleq2d AR1OnBdomR1AR1sucBA𝒫R1B
10 fvex R1BV
11 10 elpw2 A𝒫R1BAR1B
12 9 11 bitr2di AR1OnBdomR1AR1BAR1sucB
13 rankon rankAOn
14 limord LimdomR1OrddomR1
15 2 14 ax-mp OrddomR1
16 ordelon OrddomR1BdomR1BOn
17 15 16 mpan BdomR1BOn
18 17 adantl AR1OnBdomR1BOn
19 onsssuc rankAOnBOnrankABrankAsucB
20 13 18 19 sylancr AR1OnBdomR1rankABrankAsucB
21 6 12 20 3bitr4d AR1OnBdomR1AR1BrankAB