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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 3074 | . 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 3041 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-11 1469 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-in 3047 df-ss 3054 |
This theorem is referenced by: difdif2ss 3303 difdifdirss 3417 snsstp1 3640 snsstp2 3641 nnregexmid 4504 dmexg 4773 rnexg 4774 ssrnres 4951 cossxp 5031 cocnvss 5034 funinsn 5142 fabexg 5280 foimacnv 5353 ssimaex 5450 oprabss 5825 tposssxp 6114 mapsspw 6546 sbthlemi5 6817 sbthlem7 6819 caserel 6940 dmaddpi 7101 dmmulpi 7102 ltrelxr 7793 nnsscn 8689 nn0sscn 8940 nn0ssq 9376 nnssq 9377 qsscn 9379 fzval2 9748 fzossnn 9921 fzo0ssnn0 9947 expcl2lemap 10260 rpexpcl 10267 expge0 10284 expge1 10285 seq3coll 10540 summodclem2a 11105 fsum3cvg3 11120 fsumrpcl 11128 fsumge0 11183 infssuzcldc 11556 isprm3 11711 structfn 11889 strleun 11959 toponsspwpwg 12100 dmtopon 12101 lmbrf 12295 lmres 12328 txcnmpt 12353 qtopbas 12602 tgqioo 12627 dvrecap 12757 cosz12 12772 |
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