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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 4383 given excluded middle. It is an ordinal, and has some successor-like properties. For example, if 𝐴 ∈ On then both ∪ suc 𝐴 = 𝐴 (onunisuci 4444) and ∪ {𝑥 ∈ On ∣ 𝑥 ⊆ 𝐴} = 𝐴 (onuniss2 4523). Constructively (𝒫 𝐴 ∩ On) and suc 𝐴 cannot be shown to be equivalent (as proved at ordpwsucexmid 4581). (Contributed by Jim Kingdon, 21-Jul-2019.) |
Ref | Expression |
---|---|
ordpwsucss | ⊢ (Ord 𝐴 → suc 𝐴 ⊆ (𝒫 𝐴 ∩ On)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordsuc 4574 | . . . . 5 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴) | |
2 | ordelon 4395 | . . . . . 6 ⊢ ((Ord suc 𝐴 ∧ 𝑥 ∈ suc 𝐴) → 𝑥 ∈ On) | |
3 | 2 | ex 115 | . . . . 5 ⊢ (Ord suc 𝐴 → (𝑥 ∈ suc 𝐴 → 𝑥 ∈ On)) |
4 | 1, 3 | sylbi 121 | . . . 4 ⊢ (Ord 𝐴 → (𝑥 ∈ suc 𝐴 → 𝑥 ∈ On)) |
5 | ordtr 4390 | . . . . 5 ⊢ (Ord 𝐴 → Tr 𝐴) | |
6 | trsucss 4435 | . . . . 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 307 | . . 3 ⊢ (Ord 𝐴 → (𝑥 ∈ suc 𝐴 → (𝑥 ∈ On ∧ 𝑥 ⊆ 𝐴))) |
9 | elin 3330 | . . . 4 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ On) ↔ (𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 ∈ On)) | |
10 | velpw 3594 | . . . . 5 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴) | |
11 | 10 | anbi2ci 459 | . . . 4 ⊢ ((𝑥 ∈ 𝒫 𝐴 ∧ 𝑥 ∈ On) ↔ (𝑥 ∈ On ∧ 𝑥 ⊆ 𝐴)) |
12 | 9, 11 | bitri 184 | . . 3 ⊢ (𝑥 ∈ (𝒫 𝐴 ∩ On) ↔ (𝑥 ∈ On ∧ 𝑥 ⊆ 𝐴)) |
13 | 8, 12 | imbitrrdi 162 | . 2 ⊢ (Ord 𝐴 → (𝑥 ∈ suc 𝐴 → 𝑥 ∈ (𝒫 𝐴 ∩ On))) |
14 | 13 | ssrdv 3173 | 1 ⊢ (Ord 𝐴 → suc 𝐴 ⊆ (𝒫 𝐴 ∩ On)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∈ wcel 2158 ∩ cin 3140 ⊆ wss 3141 𝒫 cpw 3587 Tr wtr 4113 Ord word 4374 Oncon0 4375 suc csuc 4377 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 ax-setind 4548 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-v 2751 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-uni 3822 df-tr 4114 df-iord 4378 df-on 4380 df-suc 4383 |
This theorem is referenced by: (None) |
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