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Metamath Proof Explorer

Theorem rankr1clem

Description: Lemma for rankr1c . (Contributed by NM, 6-Oct-2003) (Revised by Mario Carneiro, 17-Nov-2014)

Ref Expression
Assertion rankr1clem ( ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) ∧ 𝐵 ∈ dom 𝑅1 ) → ( ¬ 𝐴 ∈ ( 𝑅1 ‘ 𝐵 ) ↔ 𝐵 ⊆ ( rank ‘ 𝐴 ) ) )

Proof

Step Hyp Ref Expression
1 rankr1ag ⊢ ( ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) ∧ 𝐵 ∈ dom 𝑅1 ) → ( 𝐴 ∈ ( 𝑅1 ‘ 𝐵 ) ↔ ( rank ‘ 𝐴 ) ∈ 𝐵 ) )
2 1 notbid ⊢ ( ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) ∧ 𝐵 ∈ dom 𝑅1 ) → ( ¬ 𝐴 ∈ ( 𝑅1 ‘ 𝐵 ) ↔ ¬ ( rank ‘ 𝐴 ) ∈ 𝐵 ) )
3 r1funlim ⊢ ( Fun 𝑅1 ∧ Lim dom 𝑅1 )
4 3 simpri ⊢ Lim dom 𝑅1
5 limord ⊢ ( Lim dom 𝑅1 → Ord dom 𝑅1 )
6 4 5 ax-mp ⊢ Ord dom 𝑅1
7 ordelon ⊢ ( ( Ord dom 𝑅1 ∧ 𝐵 ∈ dom 𝑅1 ) → 𝐵 ∈ On )
8 6 7 mpan ⊢ ( 𝐵 ∈ dom 𝑅1 → 𝐵 ∈ On )
9 8 adantl ⊢ ( ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) ∧ 𝐵 ∈ dom 𝑅1 ) → 𝐵 ∈ On )
10 rankon ⊢ ( rank ‘ 𝐴 ) ∈ On
11 ontri1 ⊢ ( ( 𝐵 ∈ On ∧ ( rank ‘ 𝐴 ) ∈ On ) → ( 𝐵 ⊆ ( rank ‘ 𝐴 ) ↔ ¬ ( rank ‘ 𝐴 ) ∈ 𝐵 ) )
12 9 10 11 sylancl ⊢ ( ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) ∧ 𝐵 ∈ dom 𝑅1 ) → ( 𝐵 ⊆ ( rank ‘ 𝐴 ) ↔ ¬ ( rank ‘ 𝐴 ) ∈ 𝐵 ) )
13 2 12 bitr4d ⊢ ( ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) ∧ 𝐵 ∈ dom 𝑅1 ) → ( ¬ 𝐴 ∈ ( 𝑅1 ‘ 𝐵 ) ↔ 𝐵 ⊆ ( rank ‘ 𝐴 ) ) )