![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > sstri | GIF version |
Description: Subclass transitivity inference. (Contributed by NM, 5-May-2000.) |
Ref | Expression |
---|---|
sstri.1 | ⊢ A ⊆ B |
sstri.2 | ⊢ B ⊆ 𝐶 |
Ref | Expression |
---|---|
sstri | ⊢ A ⊆ 𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sstri.1 | . 2 ⊢ A ⊆ B | |
2 | sstri.2 | . 2 ⊢ B ⊆ 𝐶 | |
3 | sstr2 2946 | . 2 ⊢ (A ⊆ B → (B ⊆ 𝐶 → A ⊆ 𝐶)) | |
4 | 1, 2, 3 | mp2 16 | 1 ⊢ A ⊆ 𝐶 |
Colors of variables: wff set class |
Syntax hints: ⊆ wss 2911 |
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-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-11 1394 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-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-in 2918 df-ss 2925 |
This theorem is referenced by: difdif2ss 3188 difdifdirss 3301 snsstp1 3505 snsstp2 3506 nnregexmid 4285 dmexg 4539 rnexg 4540 ssrnres 4706 cossxp 4786 fabexg 5020 foimacnv 5087 ssimaex 5177 oprabss 5532 tposssxp 5805 dmaddpi 6309 dmmulpi 6310 ltrelxr 6877 nnsscn 7700 nn0sscn 7962 nn0ssq 8339 nnssq 8340 qsscn 8342 fzval2 8647 fzossnn 8815 fzo0ssnn0 8841 expcl2lemap 8921 rpexpcl 8928 expge0 8945 expge1 8946 |
Copyright terms: Public domain | W3C validator |