Theorem sstrid 3148

Description: Subclass transitivity deduction. (Contributed by NM, 6-Feb-2014.)

Hypotheses

Ref	Expression
sstrid.1	⊢ 𝐴 ⊆ 𝐵
sstrid.2	⊢ (𝜑 → 𝐵 ⊆ 𝐶)

Assertion

Ref	Expression
sstrid	⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Proof of Theorem sstrid

Step	Hyp	Ref	Expression
1		sstrid.1	. . 3 ⊢ 𝐴 ⊆ 𝐵
2	1	a1i 9	. 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
3		sstrid.2	. 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐶)
4	2, 3	sstrd 3147	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Syntax hints:   → wi 4   ⊆ wss 3111

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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-11 1493  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-in 3117  df-ss 3124

This theorem is referenced by:  cossxp2  5121  fimacnv  5608  smores2  6253  f1imaen2g  6750  phplem4dom  6819  isinfinf  6854  fidcenumlemrk  6910  casef  7044  genipv  7441  fzossnn0  10100  seq3split  10404  ctinf  12300  nninfdclemcl  12320  nninfdclemp1  12322  tgcl  12605  epttop  12631  ntrin  12665  cnconst2  12774  cnrest2  12777  cnptopresti  12779  cnptoprest2  12781  hmeores  12856  blin2  12973  ivthdec  13163  limcdifap  13172  limcresi  13176  dvfgg  13198  dvcnp2cntop  13204  dvaddxxbr  13206  reeff1olem  13233