Metamath Proof Explorer


Theorem iinon

Description: The nonempty indexed intersection of a class of ordinal numbers B ( x ) is an ordinal number. (Contributed by NM, 13-Oct-2003) (Proof shortened by Mario Carneiro, 5-Dec-2016)

Ref Expression
Assertion iinon ( ( ∀ 𝑥𝐴 𝐵 ∈ On ∧ 𝐴 ≠ ∅ ) → 𝑥𝐴 𝐵 ∈ On )

Proof

Step Hyp Ref Expression
1 dfiin3g ( ∀ 𝑥𝐴 𝐵 ∈ On → 𝑥𝐴 𝐵 = ran ( 𝑥𝐴𝐵 ) )
2 1 adantr ( ( ∀ 𝑥𝐴 𝐵 ∈ On ∧ 𝐴 ≠ ∅ ) → 𝑥𝐴 𝐵 = ran ( 𝑥𝐴𝐵 ) )
3 eqid ( 𝑥𝐴𝐵 ) = ( 𝑥𝐴𝐵 )
4 3 rnmptss ( ∀ 𝑥𝐴 𝐵 ∈ On → ran ( 𝑥𝐴𝐵 ) ⊆ On )
5 dm0rn0 ( dom ( 𝑥𝐴𝐵 ) = ∅ ↔ ran ( 𝑥𝐴𝐵 ) = ∅ )
6 dmmptg ( ∀ 𝑥𝐴 𝐵 ∈ On → dom ( 𝑥𝐴𝐵 ) = 𝐴 )
7 6 eqeq1d ( ∀ 𝑥𝐴 𝐵 ∈ On → ( dom ( 𝑥𝐴𝐵 ) = ∅ ↔ 𝐴 = ∅ ) )
8 5 7 bitr3id ( ∀ 𝑥𝐴 𝐵 ∈ On → ( ran ( 𝑥𝐴𝐵 ) = ∅ ↔ 𝐴 = ∅ ) )
9 8 necon3bid ( ∀ 𝑥𝐴 𝐵 ∈ On → ( ran ( 𝑥𝐴𝐵 ) ≠ ∅ ↔ 𝐴 ≠ ∅ ) )
10 9 biimpar ( ( ∀ 𝑥𝐴 𝐵 ∈ On ∧ 𝐴 ≠ ∅ ) → ran ( 𝑥𝐴𝐵 ) ≠ ∅ )
11 oninton ( ( ran ( 𝑥𝐴𝐵 ) ⊆ On ∧ ran ( 𝑥𝐴𝐵 ) ≠ ∅ ) → ran ( 𝑥𝐴𝐵 ) ∈ On )
12 4 10 11 syl2an2r ( ( ∀ 𝑥𝐴 𝐵 ∈ On ∧ 𝐴 ≠ ∅ ) → ran ( 𝑥𝐴𝐵 ) ∈ On )
13 2 12 eqeltrd ( ( ∀ 𝑥𝐴 𝐵 ∈ On ∧ 𝐴 ≠ ∅ ) → 𝑥𝐴 𝐵 ∈ On )