Theorem psssstr 1576

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 1574	. . . . 5 |- ((A (. B /\ B (. C) -> A (. C)
2	1	exp 291	. . . 4 |- (A (. B -> (B (. C -> A (. C))
3		psseq2 1560	. . . . 5 |- (B = C -> (A (. B <-> A (. C))
4	3	biimpcd 137	. . . 4 |- (A (. B -> (B = C -> A (. C))
5	2, 4	jaod 329	. . 3 |- (A (. B -> ((B (. C \/ B = C) -> A (. C))
6	5	imp 277	. 2 |- ((A (. B /\ (B (. C \/ B = C)) -> A (. C)
7		sspss 1569	. 2 |- (B (_ C <-> (B (. C \/ B = C))
8	6, 7	sylan2b 347	1 |- ((A (. B /\ B (_ C) -> A (. C)

Syntax hints:   -> wi 2   \/ wo 195   /\ wa 196   = wceq 1091   (_ wss 1487   (. wpss 1488

This theorem is referenced by:  atexch 5769

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-in 1491  df-ss 1492  df-pss 1494

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