Theorem ordeldifsucon

Description: Membership in the difference of ordinal and successor ordinal. (Contributed by RP, 16-Jan-2025)

	Ref	Expression
Assertion	ordeldifsucon	⊢ ( ( Ord 𝐴 ∧ 𝐵 ∈ On ) → ( 𝐶 ∈ ( 𝐴 ∖ suc 𝐵 ) ↔ ( 𝐶 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶 ) ) )

Proof

Step	Hyp	Ref	Expression
1		eldif	⊢ ( 𝐶 ∈ ( 𝐴 ∖ suc 𝐵 ) ↔ ( 𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ∈ suc 𝐵 ) )
2		simplr	⊢ ( ( ( Ord 𝐴 ∧ 𝐵 ∈ On ) ∧ 𝐶 ∈ 𝐴 ) → 𝐵 ∈ On )
3		ordelord	⊢ ( ( Ord 𝐴 ∧ 𝐶 ∈ 𝐴 ) → Ord 𝐶 )
4	3	adantlr	⊢ ( ( ( Ord 𝐴 ∧ 𝐵 ∈ On ) ∧ 𝐶 ∈ 𝐴 ) → Ord 𝐶 )
5		ordelsuc	⊢ ( ( 𝐵 ∈ On ∧ Ord 𝐶 ) → ( 𝐵 ∈ 𝐶 ↔ suc 𝐵 ⊆ 𝐶 ) )
6	2 4 5	syl2anc	⊢ ( ( ( Ord 𝐴 ∧ 𝐵 ∈ On ) ∧ 𝐶 ∈ 𝐴 ) → ( 𝐵 ∈ 𝐶 ↔ suc 𝐵 ⊆ 𝐶 ) )
7		eloni	⊢ ( 𝐵 ∈ On → Ord 𝐵 )
8		ordsuci	⊢ ( Ord 𝐵 → Ord suc 𝐵 )
9	2 7 8	3syl	⊢ ( ( ( Ord 𝐴 ∧ 𝐵 ∈ On ) ∧ 𝐶 ∈ 𝐴 ) → Ord suc 𝐵 )
10		ordtri1	⊢ ( ( Ord suc 𝐵 ∧ Ord 𝐶 ) → ( suc 𝐵 ⊆ 𝐶 ↔ ¬ 𝐶 ∈ suc 𝐵 ) )
11	9 4 10	syl2anc	⊢ ( ( ( Ord 𝐴 ∧ 𝐵 ∈ On ) ∧ 𝐶 ∈ 𝐴 ) → ( suc 𝐵 ⊆ 𝐶 ↔ ¬ 𝐶 ∈ suc 𝐵 ) )
12	6 11	bitr2d	⊢ ( ( ( Ord 𝐴 ∧ 𝐵 ∈ On ) ∧ 𝐶 ∈ 𝐴 ) → ( ¬ 𝐶 ∈ suc 𝐵 ↔ 𝐵 ∈ 𝐶 ) )
13	12	pm5.32da	⊢ ( ( Ord 𝐴 ∧ 𝐵 ∈ On ) → ( ( 𝐶 ∈ 𝐴 ∧ ¬ 𝐶 ∈ suc 𝐵 ) ↔ ( 𝐶 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶 ) ) )
14	1 13	bitrid	⊢ ( ( Ord 𝐴 ∧ 𝐵 ∈ On ) → ( 𝐶 ∈ ( 𝐴 ∖ suc 𝐵 ) ↔ ( 𝐶 ∈ 𝐴 ∧ 𝐵 ∈ 𝐶 ) ) )