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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 4525. (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 4498 | . . 3 ⊢ (Ord 𝐵 → Ord suc 𝐵) | |
2 | ordelsuc 4503 | . . 3 ⊢ ((𝐴 ∈ On ∧ Ord suc 𝐵) → (𝐴 ∈ suc 𝐵 ↔ suc 𝐴 ⊆ suc 𝐵)) | |
3 | 1, 2 | sylan2 286 | . 2 ⊢ ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴 ∈ suc 𝐵 ↔ suc 𝐴 ⊆ suc 𝐵)) |
4 | ordtr 4377 | . . . 4 ⊢ (Ord 𝐵 → Tr 𝐵) | |
5 | trsucss 4422 | . . . 4 ⊢ (Tr 𝐵 → (𝐴 ∈ suc 𝐵 → 𝐴 ⊆ 𝐵)) | |
6 | 4, 5 | syl 14 | . . 3 ⊢ (Ord 𝐵 → (𝐴 ∈ suc 𝐵 → 𝐴 ⊆ 𝐵)) |
7 | 6 | adantl 277 | . 2 ⊢ ((𝐴 ∈ On ∧ Ord 𝐵) → (𝐴 ∈ suc 𝐵 → 𝐴 ⊆ 𝐵)) |
8 | 3, 7 | sylbird 170 | 1 ⊢ ((𝐴 ∈ On ∧ Ord 𝐵) → (suc 𝐴 ⊆ suc 𝐵 → 𝐴 ⊆ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∈ wcel 2148 ⊆ wss 3129 Tr wtr 4100 Ord word 4361 Oncon0 4362 suc csuc 4364 |
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-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-sn 3598 df-uni 3810 df-tr 4101 df-iord 4365 df-suc 4370 |
This theorem is referenced by: nnsucsssuc 6489 |
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