Theorem psssstrd 3054

Description: Transitivity involving subclass and proper subclass inclusion. Deduction form of psssstr 3051. (Contributed by David Moews, 1-May-2017.)

Hypotheses

Ref	Expression
psssstrd.1	|- ( ph -> A C. B )
psssstrd.2	|- ( ph -> B C_ C )

Assertion

Ref	Expression
psssstrd	|- ( ph -> A C. C )

Proof of Theorem psssstrd

Step	Hyp	Ref	Expression
1		psssstrd.1	. 2 |- ( ph -> A C. B )
2		psssstrd.2	. 2 |- ( ph -> B C_ C )
3		psssstr 3051	. 2 |- ( ( A C. B /\ B C_ C ) -> A C. C )
4	1, 2, 3	syl2anc 391	1 |- ( ph -> A C. C )

Syntax hints:   -> wi 4   C_ wss 2917   C. wpss 2918

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-in1 544  ax-in2 545  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-ne 2206  df-in 2924  df-ss 2931  df-pss 2933

This theorem is referenced by: (None)