Theorem syl6ss 2951

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 2949	1 ⊢ (φ → A ⊆ 𝐶)

Syntax hints:   → wi 4   ⊆ wss 2911

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 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-in 2918  df-ss 2925

This theorem is referenced by:  difss2  3066  sstpr  3519  rintm  3735  eqbrrdva  4448  ssxpbm  4699  ssxp1  4700  ssxp2  4701  relfld  4789  funssxp  5003  dff2  5254  fliftf  5382  1stcof  5732  2ndcof  5733  tfrlemibfn  5883  sucinc2  5965  peano5nni  7698  ioodisj  8631  fzossnn0  8801  elfzom1elp1fzo  8828  frecuzrdgfn  8879  peano5set  9399  peano5setOLD  9400