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Metamath Proof Explorer

Theorem onunel

Description: The union of two ordinals is in a third iff both of the first two are. (Contributed by Scott Fenton, 10-Sep-2024)

Ref Expression
Assertion onunel ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( ( 𝐴 ∪ 𝐵 ) ∈ 𝐶 ↔ ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) ) )

Proof

Step Hyp Ref Expression
1 ssequn1 ⊢ ( 𝐴 ⊆ 𝐵 ↔ ( 𝐴 ∪ 𝐵 ) = 𝐵 )
2 1 biimpi ⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐴 ∪ 𝐵 ) = 𝐵 )
3 2 eleq1d ⊢ ( 𝐴 ⊆ 𝐵 → ( ( 𝐴 ∪ 𝐵 ) ∈ 𝐶 ↔ 𝐵 ∈ 𝐶 ) )
4 3 adantl ⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝐴 ⊆ 𝐵 ) → ( ( 𝐴 ∪ 𝐵 ) ∈ 𝐶 ↔ 𝐵 ∈ 𝐶 ) )
5 ontr2 ⊢ ( ( 𝐴 ∈ On ∧ 𝐶 ∈ On ) → ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶 ) → 𝐴 ∈ 𝐶 ) )
6 5 3adant2 ⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶 ) → 𝐴 ∈ 𝐶 ) )
7 6 expdimp ⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝐴 ⊆ 𝐵 ) → ( 𝐵 ∈ 𝐶 → 𝐴 ∈ 𝐶 ) )
8 7 pm4.71rd ⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝐴 ⊆ 𝐵 ) → ( 𝐵 ∈ 𝐶 ↔ ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) ) )
9 4 8 bitrd ⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝐴 ⊆ 𝐵 ) → ( ( 𝐴 ∪ 𝐵 ) ∈ 𝐶 ↔ ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) ) )
10 ssequn2 ⊢ ( 𝐵 ⊆ 𝐴 ↔ ( 𝐴 ∪ 𝐵 ) = 𝐴 )
11 10 biimpi ⊢ ( 𝐵 ⊆ 𝐴 → ( 𝐴 ∪ 𝐵 ) = 𝐴 )
12 11 eleq1d ⊢ ( 𝐵 ⊆ 𝐴 → ( ( 𝐴 ∪ 𝐵 ) ∈ 𝐶 ↔ 𝐴 ∈ 𝐶 ) )
13 12 adantl ⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝐵 ⊆ 𝐴 ) → ( ( 𝐴 ∪ 𝐵 ) ∈ 𝐶 ↔ 𝐴 ∈ 𝐶 ) )
14 ontr2 ⊢ ( ( 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( ( 𝐵 ⊆ 𝐴 ∧ 𝐴 ∈ 𝐶 ) → 𝐵 ∈ 𝐶 ) )
15 14 3adant1 ⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( ( 𝐵 ⊆ 𝐴 ∧ 𝐴 ∈ 𝐶 ) → 𝐵 ∈ 𝐶 ) )
16 15 expdimp ⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝐵 ⊆ 𝐴 ) → ( 𝐴 ∈ 𝐶 → 𝐵 ∈ 𝐶 ) )
17 16 pm4.71d ⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝐵 ⊆ 𝐴 ) → ( 𝐴 ∈ 𝐶 ↔ ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) ) )
18 13 17 bitrd ⊢ ( ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) ∧ 𝐵 ⊆ 𝐴 ) → ( ( 𝐴 ∪ 𝐵 ) ∈ 𝐶 ↔ ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) ) )
19 eloni ⊢ ( 𝐴 ∈ On → Ord 𝐴 )
20 eloni ⊢ ( 𝐵 ∈ On → Ord 𝐵 )
21 ordtri2or2 ⊢ ( ( Ord 𝐴 ∧ Ord 𝐵 ) → ( 𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴 ) )
22 19 20 21 syl2an ⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ) → ( 𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴 ) )
23 22 3adant3 ⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( 𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴 ) )
24 9 18 23 mpjaodan ⊢ ( ( 𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On ) → ( ( 𝐴 ∪ 𝐵 ) ∈ 𝐶 ↔ ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) ) )