![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > trsucss | GIF version |
Description: A member of the successor of a transitive class is a subclass of it. (Contributed by NM, 4-Oct-2003.) |
Ref | Expression |
---|---|
trsucss | ⊢ (Tr A → (B ∈ suc A → B ⊆ A)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elsuci 4106 | . 2 ⊢ (B ∈ suc A → (B ∈ A ∨ B = A)) | |
2 | trss 3854 | . . 3 ⊢ (Tr A → (B ∈ A → B ⊆ A)) | |
3 | eqimss 2991 | . . . 4 ⊢ (B = A → B ⊆ A) | |
4 | 3 | a1i 9 | . . 3 ⊢ (Tr A → (B = A → B ⊆ A)) |
5 | 2, 4 | jaod 636 | . 2 ⊢ (Tr A → ((B ∈ A ∨ B = A) → B ⊆ A)) |
6 | 1, 5 | syl5 28 | 1 ⊢ (Tr A → (B ∈ suc A → B ⊆ A)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 628 = wceq 1242 ∈ wcel 1390 ⊆ 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 |
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-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: onsucsssucr 4200 ordpwsucss 4243 |
Copyright terms: Public domain | W3C validator |