Theorem sspsstr 2147

Description: Transitive law for subclass and proper subclass.

Assertion

Ref	Expression
sspsstr	|- ((A (_ B /\ B (. C) -> A (. C)

Proof of Theorem sspsstr

Step	Hyp	Ref	Expression
1		psstr 2146	. . . . 5 |- ((A (. B /\ B (. C) -> A (. C)
2	1	ex 373	. . . 4 |- (A (. B -> (B (. C -> A (. C))
3		psseq1 2131	. . . . 5 |- (A = B -> (A (. C <-> B (. C))
4	3	biimprd 154	. . . 4 |- (A = B -> (B (. C -> A (. C))
5	2, 4	jaoi 341	. . 3 |- ((A (. B \/ A = B) -> (B (. C -> A (. C))
6	5	imp 350	. 2 |- (((A (. B \/ A = B) /\ B (. C) -> A (. C)
7		sspss 2141	. 2 |- (A (_ B <-> (A (. B \/ A = B))
8	6, 7	sylanb 449	1 |- ((A (_ B /\ B (. C) -> A (. C)

Syntax hints:   -> wi 3   \/ wo 222   /\ wa 223   = wceq 954   (_ wss 2043   (. wpss 2044

This theorem is referenced by:  php 4499  ltexprlem2 5123  suplem1pr 5141

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-in 2047  df-ss 2049  df-pss 2051

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