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Mirrors > Home > ILE Home > Th. List > trsuc | GIF version |
Description: A set whose successor belongs to a transitive class also belongs. (Contributed by NM, 5-Sep-2003.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) |
Ref | Expression |
---|---|
trsuc | ⊢ ((Tr A ∧ suc B ∈ A) → B ∈ A) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sssucid 4118 | . . . . . 6 ⊢ B ⊆ suc B | |
2 | ssexg 3887 | . . . . . 6 ⊢ ((B ⊆ suc B ∧ suc B ∈ A) → B ∈ V) | |
3 | 1, 2 | mpan 400 | . . . . 5 ⊢ (suc B ∈ A → B ∈ V) |
4 | sucidg 4119 | . . . . 5 ⊢ (B ∈ V → B ∈ suc B) | |
5 | 3, 4 | syl 14 | . . . 4 ⊢ (suc B ∈ A → B ∈ suc B) |
6 | 5 | ancri 307 | . . 3 ⊢ (suc B ∈ A → (B ∈ suc B ∧ suc B ∈ A)) |
7 | trel 3852 | . . 3 ⊢ (Tr A → ((B ∈ suc B ∧ suc B ∈ A) → B ∈ A)) | |
8 | 6, 7 | syl5 28 | . 2 ⊢ (Tr A → (suc B ∈ A → B ∈ A)) |
9 | 8 | imp 115 | 1 ⊢ ((Tr A ∧ suc B ∈ A) → B ∈ A) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∈ wcel 1390 Vcvv 2551 ⊆ wss 2911 Tr wtr 3845 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 ax-sep 3866 |
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-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-sn 3373 df-uni 3572 df-tr 3846 df-suc 4074 |
This theorem is referenced by: (None) |
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