Theorem sstri 3008

Description: Subclass transitivity inference. (Contributed by NM, 5-May-2000.)

Hypotheses

Ref	Expression
sstri.1	⊢ 𝐴 ⊆ 𝐵
sstri.2	⊢ 𝐵 ⊆ 𝐶

Assertion

Ref	Expression
sstri	⊢ 𝐴 ⊆ 𝐶

Proof of Theorem sstri

Step	Hyp	Ref	Expression
1		sstri.1	. 2 ⊢ 𝐴 ⊆ 𝐵
2		sstri.2	. 2 ⊢ 𝐵 ⊆ 𝐶
3		sstr2 3006	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ⊆ 𝐶 → 𝐴 ⊆ 𝐶))
4	1, 2, 3	mp2 16	1 ⊢ 𝐴 ⊆ 𝐶

Syntax hints:   ⊆ wss 2973

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-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-in 2979  df-ss 2986

This theorem is referenced by:  difdif2ss  3221  difdifdirss  3327  snsstp1  3535  snsstp2  3536  nnregexmid  4360  dmexg  4614  rnexg  4615  ssrnres  4783  cossxp  4863  fabexg  5097  foimacnv  5164  ssimaex  5255  oprabss  5610  tposssxp  5887  dmaddpi  6515  dmmulpi  6516  ltrelxr  7173  nnsscn  8044  nn0sscn  8293  nn0ssq  8713  nnssq  8714  qsscn  8716  fzval2  9032  fzossnn  9198  fzo0ssnn0  9224  serige0  9473  expcl2lemap  9488  rpexpcl  9495  expge0  9512  expge1  9513  infssuzcldc  10347  isprm3  10500