Theorem ordtr2 3708

Description: Transitive law for ordinal classes. (The proof was shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
ordtr2	|- ((Ord A /\ Ord C) -> ((A C_ B /\ B e. C) -> A e. C))

Proof of Theorem ordtr2

Step	Hyp	Ref	Expression
1		ordelord 3680	. . . . . . . 8 |- ((Ord C /\ B e. C) -> Ord B)
2	1	ex 402	. . . . . . 7 |- (Ord C -> (B e. C -> Ord B))
3	2	ancld 322	. . . . . 6 |- (Ord C -> (B e. C -> (B e. C /\ Ord B)))
4	3	anc2li 326	. . . . 5 |- (Ord C -> (B e. C -> (Ord C /\ (B e. C /\ Ord B))))
5		ordelpss 3686	. . . . . . . . . . 11 |- ((Ord B /\ Ord C) -> (B e. C <-> B C. C))
6	5	ancoms 484	. . . . . . . . . 10 |- ((Ord C /\ Ord B) -> (B e. C <-> B C. C))
7		sspsstr 2715	. . . . . . . . . . 11 |- ((A C_ B /\ B C. C) -> A C. C)
8	7	expcom 403	. . . . . . . . . 10 |- (B C. C -> (A C_ B -> A C. C))
9	6, 8	syl6bi 231	. . . . . . . . 9 |- ((Ord C /\ Ord B) -> (B e. C -> (A C_ B -> A C. C)))
10	9	ex 402	. . . . . . . 8 |- (Ord C -> (Ord B -> (B e. C -> (A C_ B -> A C. C))))
11	10	com23 36	. . . . . . 7 |- (Ord C -> (B e. C -> (Ord B -> (A C_ B -> A C. C))))
12	11	imp32 390	. . . . . 6 |- ((Ord C /\ (B e. C /\ Ord B)) -> (A C_ B -> A C. C))
13	12	com12 14	. . . . 5 |- (A C_ B -> ((Ord C /\ (B e. C /\ Ord B)) -> A C. C))
14	4, 13	syl9 71	. . . 4 |- (Ord C -> (A C_ B -> (B e. C -> A C. C)))
15	14	imp3a 388	. . 3 |- (Ord C -> ((A C_ B /\ B e. C) -> A C. C))
16	15	adantl 424	. 2 |- ((Ord A /\ Ord C) -> ((A C_ B /\ B e. C) -> A C. C))
17		ordelpss 3686	. 2 |- ((Ord A /\ Ord C) -> (A e. C <-> A C. C))
18	16, 17	sylibrd 221	1 |- ((Ord A /\ Ord C) -> ((A C_ B /\ B e. C) -> A e. C))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   e. wcel 1300   C_ wss 2593   C. wpss 2594  Ord word 3656

This theorem is referenced by:  ontr2 3711  nnarcl 5287  axdenselem5 14023

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

Copyright terms: Public domain