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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 2952 | . 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 2917 |
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 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-11 1397 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-in 2924 df-ss 2931 |
This theorem is referenced by: difdif2ss 3194 difdifdirss 3307 snsstp1 3514 snsstp2 3515 nnregexmid 4342 dmexg 4596 rnexg 4597 ssrnres 4763 cossxp 4843 fabexg 5077 foimacnv 5144 ssimaex 5234 oprabss 5590 tposssxp 5864 dmaddpi 6423 dmmulpi 6424 ltrelxr 7080 nnsscn 7919 nn0sscn 8186 nn0ssq 8563 nnssq 8564 qsscn 8566 fzval2 8877 fzossnn 9045 fzo0ssnn0 9071 serige0 9252 expcl2lemap 9267 rpexpcl 9274 expge0 9291 expge1 9292 |
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