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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 4212. (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 4192 | . . 3 ⊢ (Ord B → Ord suc B) | |
2 | ordelsuc 4197 | . . 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 4081 | . . . 4 ⊢ (Ord B → Tr B) | |
5 | trsucss 4126 | . . . 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 1390 ⊆ wss 2911 Tr wtr 3845 Ord word 4065 Oncon0 4066 suc csuc 4068 |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-sn 3373 df-uni 3572 df-tr 3846 df-iord 4069 df-suc 4074 |
This theorem is referenced by: nnsucsssuc 6010 |
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