Theorem psstr 2146

Description: Transitive law for proper subclass. Theorem 9 of [Suppes] p. 23.

Assertion

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

Proof of Theorem psstr

Step	Hyp	Ref	Expression
1		pssss 2139	. . . 4 |- (A (. B -> A (_ B)
2		pssss 2139	. . . 4 |- (B (. C -> B (_ C)
3	1, 2	sylan9ss 2071	. . 3 |- ((A (. B /\ B (. C) -> A (_ C)
4		pssn2lp 2143	. . . . 5 |- -. (C (. B /\ B (. C)
5		psseq1 2131	. . . . . 6 |- (A = C -> (A (. B <-> C (. B))
6	5	anbi1d 616	. . . . 5 |- (A = C -> ((A (. B /\ B (. C) <-> (C (. B /\ B (. C)))
7	4, 6	mtbiri 716	. . . 4 |- (A = C -> -. (A (. B /\ B (. C))
8	7	con2i 97	. . 3 |- ((A (. B /\ B (. C) -> -. A = C)
9	3, 8	jca 288	. 2 |- ((A (. B /\ B (. C) -> (A (_ C /\ -. A = C))
10		dfpss2 2129	. 2 |- (A (. C <-> (A (_ C /\ -. A = C))
11	9, 10	sylibr 200	1 |- ((A (. B /\ B (. C) -> A (. C)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   = wceq 954   (_ wss 2043   (. wpss 2044

This theorem is referenced by:  sspsstr 2147  psssstr 2148  inf3lem5 4597  zorn 4777  ltsopr 5116

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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