Theorem ordtr2 3008

Description: Transitive law for ordinal classes.

Assertion

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

Proof of Theorem ordtr2

Step	Hyp	Ref	Expression
1		ordsseleq 2982	. . . . . . . . 9 |- ((Ord A /\ Ord B) -> (A (_ B <-> (A e. B \/ A = B)))
2	1	biimpd 153	. . . . . . . 8 |- ((Ord A /\ Ord B) -> (A (_ B -> (A e. B \/ A = B)))
3		ordtr1 3007	. . . . . . . . . . . 12 |- (Ord C -> ((A e. B /\ B e. C) -> A e. C))
4	3	exp3a 376	. . . . . . . . . . 11 |- (Ord C -> (A e. B -> (B e. C -> A e. C)))
5		eleq1a 1546	. . . . . . . . . . . . 13 |- (B e. C -> (A = B -> A e. C))
6	5	com12 11	. . . . . . . . . . . 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 426	. . . . . . . . . 10 |- (Ord C -> ((A e. B \/ A = B) -> (B e. C -> A e. C)))
9	8	com23 32	. . . . . . . . 9 |- (Ord C -> (B e. C -> ((A e. B \/ A = B) -> A e. C)))
10	9	imp 350	. . . . . . . 8 |- ((Ord C /\ B e. C) -> ((A e. B \/ A = B) -> A e. C))
11	2, 10	syl9 57	. . . . . . 7 |- ((Ord A /\ Ord B) -> ((Ord C /\ B e. C) -> (A (_ B -> A e. C)))
12	11	ex 373	. . . . . 6 |- (Ord A -> (Ord B -> ((Ord C /\ B e. C) -> (A (_ B -> A e. C))))
13		ordelord 2976	. . . . . 6 |- ((Ord C /\ B e. C) -> Ord B)
14	12, 13	syl5 21	. . . . 5 |- (Ord A -> ((Ord C /\ B e. C) -> ((Ord C /\ B e. C) -> (A (_ B -> A e. C))))
15	14	pm2.43d 65	. . . 4 |- (Ord A -> ((Ord C /\ B e. C) -> (A (_ B -> A e. C)))
16	15	expdimp 377	. . 3 |- ((Ord A /\ Ord C) -> (B e. C -> (A (_ B -> A e. C)))
17	16	com23 32	. 2 |- ((Ord A /\ Ord C) -> (A (_ B -> (B e. C -> A e. C)))
18	17	imp3a 361	1 |- ((Ord A /\ Ord C) -> ((A (_ B /\ B e. C) -> A e. C))

Syntax hints:   -> wi 3   \/ wo 222   /\ wa 223   = wceq 958   e. wcel 960   (_ wss 2050  Ord word 2953

This theorem is referenced by:  ontr2 3010  nnarcl 4238

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-11 969  ax-12 970  ax-13 971  ax-14 972  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  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957

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