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Metamath Proof Explorer

Theorem onmindif

Description: When its successor is subtracted from a class of ordinal numbers, an ordinal number is less than the minimum of the resulting subclass. (Contributed by NM, 1-Dec-2003)

Ref Expression
Assertion onmindif ( ( 𝐴 ⊆ On ∧ 𝐵 ∈ On ) → 𝐵 ∈ ∩ ( 𝐴 ∖ suc 𝐵 ) )

Proof

Step Hyp Ref Expression
1 eldif ⊢ ( 𝑥 ∈ ( 𝐴 ∖ suc 𝐵 ) ↔ ( 𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ suc 𝐵 ) )
2 ssel2 ⊢ ( ( 𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ∈ On )
3 ontri1 ⊢ ( ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) → ( 𝑥 ⊆ 𝐵 ↔ ¬ 𝐵 ∈ 𝑥 ) )
4 onsssuc ⊢ ( ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) → ( 𝑥 ⊆ 𝐵 ↔ 𝑥 ∈ suc 𝐵 ) )
5 3 4 bitr3d ⊢ ( ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) → ( ¬ 𝐵 ∈ 𝑥 ↔ 𝑥 ∈ suc 𝐵 ) )
6 5 con1bid ⊢ ( ( 𝑥 ∈ On ∧ 𝐵 ∈ On ) → ( ¬ 𝑥 ∈ suc 𝐵 ↔ 𝐵 ∈ 𝑥 ) )
7 2 6 sylan ⊢ ( ( ( 𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴 ) ∧ 𝐵 ∈ On ) → ( ¬ 𝑥 ∈ suc 𝐵 ↔ 𝐵 ∈ 𝑥 ) )
8 7 biimpd ⊢ ( ( ( 𝐴 ⊆ On ∧ 𝑥 ∈ 𝐴 ) ∧ 𝐵 ∈ On ) → ( ¬ 𝑥 ∈ suc 𝐵 → 𝐵 ∈ 𝑥 ) )
9 8 exp31 ⊢ ( 𝐴 ⊆ On → ( 𝑥 ∈ 𝐴 → ( 𝐵 ∈ On → ( ¬ 𝑥 ∈ suc 𝐵 → 𝐵 ∈ 𝑥 ) ) ) )
10 9 com23 ⊢ ( 𝐴 ⊆ On → ( 𝐵 ∈ On → ( 𝑥 ∈ 𝐴 → ( ¬ 𝑥 ∈ suc 𝐵 → 𝐵 ∈ 𝑥 ) ) ) )
11 10 imp4b ⊢ ( ( 𝐴 ⊆ On ∧ 𝐵 ∈ On ) → ( ( 𝑥 ∈ 𝐴 ∧ ¬ 𝑥 ∈ suc 𝐵 ) → 𝐵 ∈ 𝑥 ) )
12 1 11 biimtrid ⊢ ( ( 𝐴 ⊆ On ∧ 𝐵 ∈ On ) → ( 𝑥 ∈ ( 𝐴 ∖ suc 𝐵 ) → 𝐵 ∈ 𝑥 ) )
13 12 ralrimiv ⊢ ( ( 𝐴 ⊆ On ∧ 𝐵 ∈ On ) → ∀ 𝑥 ∈ ( 𝐴 ∖ suc 𝐵 ) 𝐵 ∈ 𝑥 )
14 elintg ⊢ ( 𝐵 ∈ On → ( 𝐵 ∈ ∩ ( 𝐴 ∖ suc 𝐵 ) ↔ ∀ 𝑥 ∈ ( 𝐴 ∖ suc 𝐵 ) 𝐵 ∈ 𝑥 ) )
15 14 adantl ⊢ ( ( 𝐴 ⊆ On ∧ 𝐵 ∈ On ) → ( 𝐵 ∈ ∩ ( 𝐴 ∖ suc 𝐵 ) ↔ ∀ 𝑥 ∈ ( 𝐴 ∖ suc 𝐵 ) 𝐵 ∈ 𝑥 ) )
16 13 15 mpbird ⊢ ( ( 𝐴 ⊆ On ∧ 𝐵 ∈ On ) → 𝐵 ∈ ∩ ( 𝐴 ∖ suc 𝐵 ) )