Metamath Proof Explorer


Theorem onsuct0

Description: A successor ordinal number is a T_0 space. (Contributed by Chen-Pang He, 8-Nov-2015)

Ref Expression
Assertion onsuct0 ( 𝐴 ∈ On → suc 𝐴 ∈ Kol2 )

Proof

Step Hyp Ref Expression
1 eloni ( 𝐴 ∈ On → Ord 𝐴 )
2 df-ral ( ∀ 𝑜 ∈ suc 𝐴 ( 𝑥𝑜𝑦𝑜 ) ↔ ∀ 𝑜 ( 𝑜 ∈ suc 𝐴 → ( 𝑥𝑜𝑦𝑜 ) ) )
3 ordelon ( ( Ord 𝐴𝑥𝐴 ) → 𝑥 ∈ On )
4 ordelon ( ( Ord 𝐴𝑦𝐴 ) → 𝑦 ∈ On )
5 3 4 anim12dan ( ( Ord 𝐴 ∧ ( 𝑥𝐴𝑦𝐴 ) ) → ( 𝑥 ∈ On ∧ 𝑦 ∈ On ) )
6 ordsuc ( Ord 𝐴 ↔ Ord suc 𝐴 )
7 ordelon ( ( Ord suc 𝐴𝑜 ∈ suc 𝐴 ) → 𝑜 ∈ On )
8 7 ex ( Ord suc 𝐴 → ( 𝑜 ∈ suc 𝐴𝑜 ∈ On ) )
9 6 8 sylbi ( Ord 𝐴 → ( 𝑜 ∈ suc 𝐴𝑜 ∈ On ) )
10 9 adantr ( ( Ord 𝐴 ∧ ( 𝑥𝐴𝑦𝐴 ) ) → ( 𝑜 ∈ suc 𝐴𝑜 ∈ On ) )
11 notbi ( ( 𝑥𝑜𝑦𝑜 ) ↔ ( ¬ 𝑥𝑜 ↔ ¬ 𝑦𝑜 ) )
12 ontri1 ( ( 𝑜 ∈ On ∧ 𝑥 ∈ On ) → ( 𝑜𝑥 ↔ ¬ 𝑥𝑜 ) )
13 onsssuc ( ( 𝑜 ∈ On ∧ 𝑥 ∈ On ) → ( 𝑜𝑥𝑜 ∈ suc 𝑥 ) )
14 12 13 bitr3d ( ( 𝑜 ∈ On ∧ 𝑥 ∈ On ) → ( ¬ 𝑥𝑜𝑜 ∈ suc 𝑥 ) )
15 14 adantrr ( ( 𝑜 ∈ On ∧ ( 𝑥 ∈ On ∧ 𝑦 ∈ On ) ) → ( ¬ 𝑥𝑜𝑜 ∈ suc 𝑥 ) )
16 ontri1 ( ( 𝑜 ∈ On ∧ 𝑦 ∈ On ) → ( 𝑜𝑦 ↔ ¬ 𝑦𝑜 ) )
17 onsssuc ( ( 𝑜 ∈ On ∧ 𝑦 ∈ On ) → ( 𝑜𝑦𝑜 ∈ suc 𝑦 ) )
18 16 17 bitr3d ( ( 𝑜 ∈ On ∧ 𝑦 ∈ On ) → ( ¬ 𝑦𝑜𝑜 ∈ suc 𝑦 ) )
19 18 adantrl ( ( 𝑜 ∈ On ∧ ( 𝑥 ∈ On ∧ 𝑦 ∈ On ) ) → ( ¬ 𝑦𝑜𝑜 ∈ suc 𝑦 ) )
20 15 19 bibi12d ( ( 𝑜 ∈ On ∧ ( 𝑥 ∈ On ∧ 𝑦 ∈ On ) ) → ( ( ¬ 𝑥𝑜 ↔ ¬ 𝑦𝑜 ) ↔ ( 𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦 ) ) )
21 20 ancoms ( ( ( 𝑥 ∈ On ∧ 𝑦 ∈ On ) ∧ 𝑜 ∈ On ) → ( ( ¬ 𝑥𝑜 ↔ ¬ 𝑦𝑜 ) ↔ ( 𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦 ) ) )
22 11 21 syl5bb ( ( ( 𝑥 ∈ On ∧ 𝑦 ∈ On ) ∧ 𝑜 ∈ On ) → ( ( 𝑥𝑜𝑦𝑜 ) ↔ ( 𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦 ) ) )
23 22 biimpd ( ( ( 𝑥 ∈ On ∧ 𝑦 ∈ On ) ∧ 𝑜 ∈ On ) → ( ( 𝑥𝑜𝑦𝑜 ) → ( 𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦 ) ) )
24 5 10 23 syl6an ( ( Ord 𝐴 ∧ ( 𝑥𝐴𝑦𝐴 ) ) → ( 𝑜 ∈ suc 𝐴 → ( ( 𝑥𝑜𝑦𝑜 ) → ( 𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦 ) ) ) )
25 24 a2d ( ( Ord 𝐴 ∧ ( 𝑥𝐴𝑦𝐴 ) ) → ( ( 𝑜 ∈ suc 𝐴 → ( 𝑥𝑜𝑦𝑜 ) ) → ( 𝑜 ∈ suc 𝐴 → ( 𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦 ) ) ) )
26 ordelss ( ( Ord 𝐴𝑥𝐴 ) → 𝑥𝐴 )
27 ordelord ( ( Ord 𝐴𝑥𝐴 ) → Ord 𝑥 )
28 ordsucsssuc ( ( Ord 𝑥 ∧ Ord 𝐴 ) → ( 𝑥𝐴 ↔ suc 𝑥 ⊆ suc 𝐴 ) )
29 28 ancoms ( ( Ord 𝐴 ∧ Ord 𝑥 ) → ( 𝑥𝐴 ↔ suc 𝑥 ⊆ suc 𝐴 ) )
30 27 29 syldan ( ( Ord 𝐴𝑥𝐴 ) → ( 𝑥𝐴 ↔ suc 𝑥 ⊆ suc 𝐴 ) )
31 26 30 mpbid ( ( Ord 𝐴𝑥𝐴 ) → suc 𝑥 ⊆ suc 𝐴 )
32 31 ssneld ( ( Ord 𝐴𝑥𝐴 ) → ( ¬ 𝑜 ∈ suc 𝐴 → ¬ 𝑜 ∈ suc 𝑥 ) )
33 32 adantrr ( ( Ord 𝐴 ∧ ( 𝑥𝐴𝑦𝐴 ) ) → ( ¬ 𝑜 ∈ suc 𝐴 → ¬ 𝑜 ∈ suc 𝑥 ) )
34 ordelss ( ( Ord 𝐴𝑦𝐴 ) → 𝑦𝐴 )
35 ordelord ( ( Ord 𝐴𝑦𝐴 ) → Ord 𝑦 )
36 ordsucsssuc ( ( Ord 𝑦 ∧ Ord 𝐴 ) → ( 𝑦𝐴 ↔ suc 𝑦 ⊆ suc 𝐴 ) )
37 36 ancoms ( ( Ord 𝐴 ∧ Ord 𝑦 ) → ( 𝑦𝐴 ↔ suc 𝑦 ⊆ suc 𝐴 ) )
38 35 37 syldan ( ( Ord 𝐴𝑦𝐴 ) → ( 𝑦𝐴 ↔ suc 𝑦 ⊆ suc 𝐴 ) )
39 34 38 mpbid ( ( Ord 𝐴𝑦𝐴 ) → suc 𝑦 ⊆ suc 𝐴 )
40 39 ssneld ( ( Ord 𝐴𝑦𝐴 ) → ( ¬ 𝑜 ∈ suc 𝐴 → ¬ 𝑜 ∈ suc 𝑦 ) )
41 40 adantrl ( ( Ord 𝐴 ∧ ( 𝑥𝐴𝑦𝐴 ) ) → ( ¬ 𝑜 ∈ suc 𝐴 → ¬ 𝑜 ∈ suc 𝑦 ) )
42 33 41 jcad ( ( Ord 𝐴 ∧ ( 𝑥𝐴𝑦𝐴 ) ) → ( ¬ 𝑜 ∈ suc 𝐴 → ( ¬ 𝑜 ∈ suc 𝑥 ∧ ¬ 𝑜 ∈ suc 𝑦 ) ) )
43 pm5.21 ( ( ¬ 𝑜 ∈ suc 𝑥 ∧ ¬ 𝑜 ∈ suc 𝑦 ) → ( 𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦 ) )
44 42 43 syl6 ( ( Ord 𝐴 ∧ ( 𝑥𝐴𝑦𝐴 ) ) → ( ¬ 𝑜 ∈ suc 𝐴 → ( 𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦 ) ) )
45 idd ( ( Ord 𝐴 ∧ ( 𝑥𝐴𝑦𝐴 ) ) → ( ( 𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦 ) → ( 𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦 ) ) )
46 44 45 jad ( ( Ord 𝐴 ∧ ( 𝑥𝐴𝑦𝐴 ) ) → ( ( 𝑜 ∈ suc 𝐴 → ( 𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦 ) ) → ( 𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦 ) ) )
47 25 46 syld ( ( Ord 𝐴 ∧ ( 𝑥𝐴𝑦𝐴 ) ) → ( ( 𝑜 ∈ suc 𝐴 → ( 𝑥𝑜𝑦𝑜 ) ) → ( 𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦 ) ) )
48 47 alimdv ( ( Ord 𝐴 ∧ ( 𝑥𝐴𝑦𝐴 ) ) → ( ∀ 𝑜 ( 𝑜 ∈ suc 𝐴 → ( 𝑥𝑜𝑦𝑜 ) ) → ∀ 𝑜 ( 𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦 ) ) )
49 2 48 syl5bi ( ( Ord 𝐴 ∧ ( 𝑥𝐴𝑦𝐴 ) ) → ( ∀ 𝑜 ∈ suc 𝐴 ( 𝑥𝑜𝑦𝑜 ) → ∀ 𝑜 ( 𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦 ) ) )
50 dfcleq ( suc 𝑥 = suc 𝑦 ↔ ∀ 𝑜 ( 𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦 ) )
51 suc11 ( ( 𝑥 ∈ On ∧ 𝑦 ∈ On ) → ( suc 𝑥 = suc 𝑦𝑥 = 𝑦 ) )
52 50 51 bitr3id ( ( 𝑥 ∈ On ∧ 𝑦 ∈ On ) → ( ∀ 𝑜 ( 𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦 ) ↔ 𝑥 = 𝑦 ) )
53 5 52 syl ( ( Ord 𝐴 ∧ ( 𝑥𝐴𝑦𝐴 ) ) → ( ∀ 𝑜 ( 𝑜 ∈ suc 𝑥𝑜 ∈ suc 𝑦 ) ↔ 𝑥 = 𝑦 ) )
54 49 53 sylibd ( ( Ord 𝐴 ∧ ( 𝑥𝐴𝑦𝐴 ) ) → ( ∀ 𝑜 ∈ suc 𝐴 ( 𝑥𝑜𝑦𝑜 ) → 𝑥 = 𝑦 ) )
55 54 ralrimivva ( Ord 𝐴 → ∀ 𝑥𝐴𝑦𝐴 ( ∀ 𝑜 ∈ suc 𝐴 ( 𝑥𝑜𝑦𝑜 ) → 𝑥 = 𝑦 ) )
56 1 55 syl ( 𝐴 ∈ On → ∀ 𝑥𝐴𝑦𝐴 ( ∀ 𝑜 ∈ suc 𝐴 ( 𝑥𝑜𝑦𝑜 ) → 𝑥 = 𝑦 ) )
57 onsuctopon ( 𝐴 ∈ On → suc 𝐴 ∈ ( TopOn ‘ 𝐴 ) )
58 ist0-2 ( suc 𝐴 ∈ ( TopOn ‘ 𝐴 ) → ( suc 𝐴 ∈ Kol2 ↔ ∀ 𝑥𝐴𝑦𝐴 ( ∀ 𝑜 ∈ suc 𝐴 ( 𝑥𝑜𝑦𝑜 ) → 𝑥 = 𝑦 ) ) )
59 57 58 syl ( 𝐴 ∈ On → ( suc 𝐴 ∈ Kol2 ↔ ∀ 𝑥𝐴𝑦𝐴 ( ∀ 𝑜 ∈ suc 𝐴 ( 𝑥𝑜𝑦𝑜 ) → 𝑥 = 𝑦 ) ) )
60 56 59 mpbird ( 𝐴 ∈ On → suc 𝐴 ∈ Kol2 )