Theorem sstri 2954

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 2952	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ⊆ 𝐶 → 𝐴 ⊆ 𝐶))
4	1, 2, 3	mp2 16	1 ⊢ 𝐴 ⊆ 𝐶

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