Theorem ordtr2OLD 3709

Description: Transitive law for ordinal classes.

Assertion

Ref	Expression
ordtr2OLD	|- ((Ord A /\ Ord C) -> ((A C_ B /\ B e. C) -> A e. C))

Proof of Theorem ordtr2OLD

Step	Hyp	Ref	Expression
1		ordsseleq 3687	. . . . . . . . 9 |- ((Ord A /\ Ord B) -> (A C_ B <-> (A e. B \/ A = B)))
2	1	biimpd 170	. . . . . . . 8 |- ((Ord A /\ Ord B) -> (A C_ B -> (A e. B \/ A = B)))
3		ordtr1 3707	. . . . . . . . . . . 12 |- (Ord C -> ((A e. B /\ B e. C) -> A e. C))
4	3	exp3a 405	. . . . . . . . . . 11 |- (Ord C -> (A e. B -> (B e. C -> A e. C)))
5		eleq1a 1966	. . . . . . . . . . . . 13 |- (B e. C -> (A = B -> A e. C))
6	5	com12 14	. . . . . . . . . . . 12 |- (A = B -> (B e. C -> A e. C))
7	6	a1i 8	. . . . . . . . . . 11 |- (Ord C -> (A = B -> (B e. C -> A e. C)))
8	4, 7	jaod 469	. . . . . . . . . 10 |- (Ord C -> ((A e. B \/ A = B) -> (B e. C -> A e. C)))
9	8	com23 36	. . . . . . . . 9 |- (Ord C -> (B e. C -> ((A e. B \/ A = B) -> A e. C)))
10	9	imp 377	. . . . . . . 8 |- ((Ord C /\ B e. C) -> ((A e. B \/ A = B) -> A e. C))
11	2, 10	syl9 71	. . . . . . 7 |- ((Ord A /\ Ord B) -> ((Ord C /\ B e. C) -> (A C_ B -> A e. C)))
12	11	ex 402	. . . . . 6 |- (Ord A -> (Ord B -> ((Ord C /\ B e. C) -> (A C_ B -> A e. C))))
13		ordelord 3680	. . . . . 6 |- ((Ord C /\ B e. C) -> Ord B)
14	12, 13	syl5 20	. . . . 5 |- (Ord A -> ((Ord C /\ B e. C) -> ((Ord C /\ B e. C) -> (A C_ B -> A e. C))))
15	14	pm2.43d 79	. . . 4 |- (Ord A -> ((Ord C /\ B e. C) -> (A C_ B -> A e. C)))
16	15	expdimp 406	. . 3 |- ((Ord A /\ Ord C) -> (B e. C -> (A C_ B -> A e. C)))
17	16	com23 36	. 2 |- ((Ord A /\ Ord C) -> (A C_ B -> (B e. C -> A e. C)))
18	17	imp3a 388	1 |- ((Ord A /\ Ord C) -> ((A C_ B /\ B e. C) -> A e. C))

Syntax hints:   -> wi 3   \/ wo 239   /\ wa 240   = wceq 1298   e. wcel 1300   C_ wss 2593  Ord word 3656

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-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  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  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660

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