Theorem sstri 2948

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 2946	. 2 ⊢ (A ⊆ B → (B ⊆ 𝐶 → A ⊆ 𝐶))
4	1, 2, 3	mp2 16	1 ⊢ A ⊆ 𝐶

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  6837  nnsscn  7660  nn0sscn  7922  nn0ssq  8299  nnssq  8300  qsscn  8302  fzval2  8607  fzossnn  8775  fzo0ssnn0  8801  expcl2lemap  8881  rpexpcl  8888  expge0  8905  expge1  8906