Theorem syl5ss 3010

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 3009	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Syntax hints:   → wi 4   ⊆ 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:  fimacnv  5317  smores2  5932  f1imaen2g  6296  phplem4dom  6348  genipv  6699  fzossnn0  9184