Theorem psssstr 2155

Description: Transitive law for subclass and proper subclass.

Assertion

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

Proof of Theorem psssstr

Step	Hyp	Ref	Expression
1		psstr 2153	. . . . 5 |- ((A (. B /\ B (. C) -> A (. C)
2	1	ex 373	. . . 4 |- (A (. B -> (B (. C -> A (. C))
3		psseq2 2139	. . . . 5 |- (B = C -> (A (. B <-> A (. C))
4	3	biimpcd 155	. . . 4 |- (A (. B -> (B = C -> A (. C))
5	2, 4	jaod 426	. . 3 |- (A (. B -> ((B (. C \/ B = C) -> A (. C))
6	5	imp 350	. 2 |- ((A (. B /\ (B (. C \/ B = C)) -> A (. C)
7		sspss 2148	. 2 |- (B (_ C <-> (B (. C \/ B = C))
8	6, 7	sylan2b 454	1 |- ((A (. B /\ B (_ C) -> A (. C)

Syntax hints:   -> wi 3   \/ wo 222   /\ wa 223   = wceq 958   (_ wss 2050   (. wpss 2051

This theorem is referenced by:  atexcht 10303

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-in 2054  df-ss 2056  df-pss 2058

Copyright terms: Public domain