Theorem syl5ss 3058

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

Hypotheses

Ref	Expression
syl5ss.1	⊢ 𝐴 ⊆ 𝐵
syl5ss.2	⊢ (𝜑 → 𝐵 ⊆ 𝐶)

Assertion

Ref	Expression
syl5ss	⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Proof of Theorem syl5ss

Step	Hyp	Ref	Expression
1		syl5ss.1	. . 3 ⊢ 𝐴 ⊆ 𝐵
2	1	a1i 9	. 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
3		syl5ss.2	. 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐶)
4	2, 3	sstrd 3057	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Syntax hints:   → wi 4   ⊆ wss 3021

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-11 1452  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-in 3027  df-ss 3034

This theorem is referenced by:  cossxp2  4998  fimacnv  5481  smores2  6121  f1imaen2g  6617  phplem4dom  6685  isinfinf  6720  fidcenumlemrk  6770  casef  6888  genipv  7218  fzossnn0  9793  seq3split  10093  ctinf  11735  tgcl  12015  epttop  12041  ntrin  12075  cnconst2  12183  cnrest2  12186  cnptopresti  12188  cnptoprest2  12190  blin2  12360  limcdifap  12512  limcresi  12515  dvfgg  12530