Theorem syl6ss 2957

Description: Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)

Hypotheses

Ref	Expression
syl6ss.1	⊢ (𝜑 → 𝐴 ⊆ 𝐵)
syl6ss.2	⊢ 𝐵 ⊆ 𝐶

Assertion

Ref	Expression
syl6ss	⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Proof of Theorem syl6ss

Step	Hyp	Ref	Expression
1		syl6ss.1	. 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
2		syl6ss.2	. . 3 ⊢ 𝐵 ⊆ 𝐶
3	2	a1i 9	. 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐶)
4	1, 3	sstrd 2955	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Syntax hints:   → wi 4   ⊆ wss 2917

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 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-in 2924  df-ss 2931

This theorem is referenced by:  difss2  3072  sstpr  3528  rintm  3744  eqbrrdva  4505  ssxpbm  4756  ssxp1  4757  ssxp2  4758  relfld  4846  funssxp  5060  dff2  5311  fliftf  5439  1stcof  5790  2ndcof  5791  tfrlemibfn  5942  sucinc2  6026  peano5nnnn  6966  peano5nni  7917  ioodisj  8861  fzossnn0  9031  elfzom1elp1fzo  9058  frecuzrdgfn  9198  peano5set  10064  peano5setOLD  10065