Theorem sylan9ss 3208

Description: A subclass transitivity deduction. (Contributed by NM, 27-Sep-2004.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Hypotheses

Ref	Expression
sylan9ss.1	⊢ (𝜑 → 𝐴 ⊆ 𝐵)
sylan9ss.2	⊢ (𝜓 → 𝐵 ⊆ 𝐶)

Assertion

Ref	Expression
sylan9ss	⊢ ((𝜑 ∧ 𝜓) → 𝐴 ⊆ 𝐶)

Proof of Theorem sylan9ss

Step	Hyp	Ref	Expression
1		sylan9ss.1	. 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
2		sylan9ss.2	. 2 ⊢ (𝜓 → 𝐵 ⊆ 𝐶)
3		sstr 3203	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊆ 𝐶)
4	1, 2, 3	syl2an 289	1 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ⊆ 𝐶)

Syntax hints:   → wi 4   ∧ wa 104   ⊆ wss 3168

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-in 3174  df-ss 3181

This theorem is referenced by:  sylan9ssr  3209  unss12  3347  ss2in  3403  relrelss  5215  funssxp  5452