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Mirrors > Home > ILE Home > Th. List > onsucsssucr | GIF version |
Description: The subclass relationship between two ordinals is inherited by their predecessors. The converse implies excluded middle, as shown at onsucsssucexmid 4252. (Contributed by Mario Carneiro and Jim Kingdon, 29-Jul-2019.) |
Ref | Expression |
---|---|
onsucsssucr | ⊢ ((𝐴 ∈ On ∧ Ord 𝐵) → (suc 𝐴 ⊆ suc 𝐵 → 𝐴 ⊆ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordsucim 4226 | . . 3 ⊢ (Ord 𝐵 → Ord suc 𝐵) | |
2 | ordelsuc 4231 | . . 3 ⊢ ((𝐴 ∈ On ∧ Ord suc 𝐵) → (𝐴 ∈ suc 𝐵 ↔ suc 𝐴 ⊆ suc 𝐵)) | |
3 | 1, 2 | sylan2 270 | . 2 ⊢ ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴 ∈ suc 𝐵 ↔ suc 𝐴 ⊆ suc 𝐵)) |
4 | ordtr 4115 | . . . 4 ⊢ (Ord 𝐵 → Tr 𝐵) | |
5 | trsucss 4160 | . . . 4 ⊢ (Tr 𝐵 → (𝐴 ∈ suc 𝐵 → 𝐴 ⊆ 𝐵)) | |
6 | 4, 5 | syl 14 | . . 3 ⊢ (Ord 𝐵 → (𝐴 ∈ suc 𝐵 → 𝐴 ⊆ 𝐵)) |
7 | 6 | adantl 262 | . 2 ⊢ ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴 ∈ suc 𝐵 → 𝐴 ⊆ 𝐵)) |
8 | 3, 7 | sylbird 159 | 1 ⊢ ((𝐴 ∈ On ∧ Ord 𝐵) → (suc 𝐴 ⊆ suc 𝐵 → 𝐴 ⊆ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∈ wcel 1393 ⊆ wss 2917 Tr wtr 3854 Ord word 4099 Oncon0 4100 suc csuc 4102 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-sn 3381 df-uni 3581 df-tr 3855 df-iord 4103 df-suc 4108 |
This theorem is referenced by: nnsucsssuc 6071 |
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