Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > ordpwsucss | GIF version |
Description: The collection of
ordinals in the power class of an ordinal is a
superset of its successor.
We can think of (𝒫 𝐴 ∩ On) as another possible definition of successor, which would be equivalent to df-suc 4288 given excluded middle. It is an ordinal, and has some successor-like properties. For example, if 𝐴 ∈ On then both ∪ suc 𝐴 = 𝐴 (onunisuci 4349) and ∪ {𝑥 ∈ On ∣ 𝑥 ⊆ 𝐴} = 𝐴 (onuniss2 4423). Constructively (𝒫 𝐴 ∩ On) and suc 𝐴 cannot be shown to be equivalent (as proved at ordpwsucexmid 4480). (Contributed by Jim Kingdon, 21-Jul-2019.) |
Ref | Expression |
---|---|
ordpwsucss | ⊢ (Ord 𝐴 → suc 𝐴 ⊆ (𝒫 𝐴 ∩ On)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordsuc 4473 | . . . . 5 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴) | |
2 | ordelon 4300 | . . . . . 6 ⊢ ((Ord suc 𝐴 ∧ 𝑥 ∈ suc 𝐴) → 𝑥 ∈ On) | |
3 | 2 | ex 114 | . . . . 5 ⊢ (Ord suc 𝐴 → (𝑥 ∈ suc 𝐴 → 𝑥 ∈ On)) |
4 | 1, 3 | sylbi 120 | . . . 4 ⊢ (Ord 𝐴 → (𝑥 ∈ suc 𝐴 → 𝑥 ∈ On)) |
5 | ordtr 4295 | . . . . 5 ⊢ (Ord 𝐴 → Tr 𝐴) | |
6 | trsucss 4340 | . . . . 5 ⊢ (Tr 𝐴 → (𝑥 ∈ suc 𝐴 → 𝑥 ⊆ 𝐴)) | |
7 | 5, 6 | syl 14 | . . . 4 ⊢ (Ord 𝐴 → (𝑥 ∈ suc 𝐴 → 𝑥 ⊆ 𝐴)) |
8 | 4, 7 | jcad 305 | . . 3 ⊢ (Ord 𝐴 → (𝑥 ∈ suc 𝐴 → (𝑥 ∈ On ∧ 𝑥 ⊆ 𝐴))) |
9 | elin 3254 | . . . 4 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ On) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 ∈ On)) | |
10 | velpw 3512 | . . . . 5 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) | |
11 | 10 | anbi2ci 454 | . . . 4 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 ∈ On) ↔ (𝑥 ∈ On ∧ 𝑥 ⊆ 𝐴)) |
12 | 9, 11 | bitri 183 | . . 3 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ On) ↔ (𝑥 ∈ On ∧ 𝑥 ⊆ 𝐴)) |
13 | 8, 12 | syl6ibr 161 | . 2 ⊢ (Ord 𝐴 → (𝑥 ∈ suc 𝐴 → 𝑥 ∈ (𝒫 𝐴 ∩ On))) |
14 | 13 | ssrdv 3098 | 1 ⊢ (Ord 𝐴 → suc 𝐴 ⊆ (𝒫 𝐴 ∩ On)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1480 ∩ cin 3065 ⊆ wss 3066 𝒫 cpw 3505 Tr wtr 4021 Ord word 4279 Oncon0 4280 suc csuc 4282 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-setind 4447 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-uni 3732 df-tr 4022 df-iord 4283 df-on 4285 df-suc 4288 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |