Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 4170 | . . . . . 6 ⊢ 𝐵 ⊆ suc 𝐵 | |
2 | ssexg 3917 | . . . . . 6 ⊢ ((𝐵 ⊆ suc 𝐵 ∧ suc 𝐵 ∈ 𝐴) → 𝐵 ∈ V) | |
3 | 1, 2 | mpan 414 | . . . . 5 ⊢ (suc 𝐵 ∈ 𝐴 → 𝐵 ∈ V) |
4 | sucidg 4171 | . . . . 5 ⊢ (𝐵 ∈ V → 𝐵 ∈ suc 𝐵) | |
5 | 3, 4 | syl 14 | . . . 4 ⊢ (suc 𝐵 ∈ 𝐴 → 𝐵 ∈ suc 𝐵) |
6 | 5 | ancri 317 | . . 3 ⊢ (suc 𝐵 ∈ 𝐴 → (𝐵 ∈ suc 𝐵 ∧ suc 𝐵 ∈ 𝐴)) |
7 | trel 3882 | . . 3 ⊢ (Tr 𝐴 → ((𝐵 ∈ suc 𝐵 ∧ suc 𝐵 ∈ 𝐴) → 𝐵 ∈ 𝐴)) | |
8 | 6, 7 | syl5 32 | . 2 ⊢ (Tr 𝐴 → (suc 𝐵 ∈ 𝐴 → 𝐵 ∈ 𝐴)) |
9 | 8 | imp 122 | 1 ⊢ ((Tr 𝐴 ∧ suc 𝐵 ∈ 𝐴) → 𝐵 ∈ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ∈ wcel 1433 Vcvv 2601 ⊆ wss 2973 Tr wtr 3875 suc csuc 4120 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 |
This theorem depends on definitions: df-bi 115 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-sn 3404 df-uni 3602 df-tr 3876 df-suc 4126 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |