Theorem sstrALT2 16659

Description: Virtual deduction proof of sstr 2625, transitivity of subclasses, Theorem 6 of [Suppes] p. 23. This theorem was automatically generated from sstrALT2VD 16658 using the command file translate_without_overwriting.cmd . It was not minimized because the automated minimization excluding duplicates generates a minimized proof which, although not directly containing any duplicates, indirectly contains a duplicate. That is, the trace back of the minimized proof contains a duplicate. This is undesirable because some step(s) of the minimized proof use the proven theorem.

Assertion

Ref	Expression
sstrALT2	|- ((A C_ B /\ B C_ C) -> A C_ C)

Proof of Theorem sstrALT2

Step	Hyp	Ref	Expression
1		dfss2 2610	. . 3 |- (A C_ C <-> A.x(x e. A -> x e. C))
2		id 73	. . . . . . 7 |- ((A C_ B /\ B C_ C) -> (A C_ B /\ B C_ C))
3		simpr 350	. . . . . . 7 |- ((A C_ B /\ B C_ C) -> B C_ C)
4	2, 3	syl 12	. . . . . 6 |- ((A C_ B /\ B C_ C) -> B C_ C)
5		simpl 346	. . . . . . . 8 |- ((A C_ B /\ B C_ C) -> A C_ B)
6	2, 5	syl 12	. . . . . . 7 |- ((A C_ B /\ B C_ C) -> A C_ B)
7		idd 75	. . . . . . 7 |- ((A C_ B /\ B C_ C) -> (x e. A -> x e. A))
8		ssel2 2616	. . . . . . 7 |- ((A C_ B /\ x e. A) -> x e. B)
9	6, 7, 8	ee12an 1273	. . . . . 6 |- ((A C_ B /\ B C_ C) -> (x e. A -> x e. B))
10		ssel2 2616	. . . . . 6 |- ((B C_ C /\ x e. B) -> x e. C)
11	4, 9, 10	ee12an 1273	. . . . 5 |- ((A C_ B /\ B C_ C) -> (x e. A -> x e. C))
12	11	iin2 16507	. . . 4 |- ((A C_ B /\ B C_ C) -> (x e. A -> x e. C))
13	12	19.21aiv 1664	. . 3 |- ((A C_ B /\ B C_ C) -> A.x(x e. A -> x e. C))
14		bi2 166	. . 3 |- ((A C_ C <-> A.x(x e. A -> x e. C)) -> (A.x(x e. A -> x e. C) -> A C_ C))
15	1, 13, 14	ee01 16582	. 2 |- ((A C_ B /\ B C_ C) -> A C_ C)
16	15	iin1 16482	1 |- ((A C_ B /\ B C_ C) -> A C_ C)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   e. wcel 1300   C_ wss 2593

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-10 1308  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-in 2603  df-ss 2605

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