Theorem ordtr2 3002

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 2976	. . . . . . . . . 10 |- ((Ord A /\ Ord B) -> (A (_ B <-> (A e. B \/ A = B)))
2	1	biimpd 153	. . . . . . . . 9 |- ((Ord A /\ Ord B) -> (A (_ B -> (A e. B \/ A = B)))
3		ordtr1 3001	. . . . . . . . . . . . 13 |- (Ord C -> ((A e. B /\ B e. C) -> A e. C))
4	3	exp3a 375	. . . . . . . . . . . 12 |- (Ord C -> (A e. B -> (B e. C -> A e. C)))
5		eleq1a 1543	. . . . . . . . . . . . . 14 |- (B e. C -> (A = B -> A e. C))
6	5	com12 11	. . . . . . . . . . . . 13 |- (A = B -> (B e. C -> A e. C))
7	6	a1i 8	. . . . . . . . . . . 12 |- (Ord C -> (A = B -> (B e. C -> A e. C)))
8	4, 7	jaod 424	. . . . . . . . . . 11 |- (Ord C -> ((A e. B \/ A = B) -> (B e. C -> A e. C)))
9	8	com23 32	. . . . . . . . . 10 |- (Ord C -> (B e. C -> ((A e. B \/ A = B) -> A e. C)))
10	9	imp 350	. . . . . . . . 9 |- ((Ord C /\ B e. C) -> ((A e. B \/ A = B) -> A e. C))
11	2, 10	syl9 57	. . . . . . . 8 |- ((Ord A /\ Ord B) -> ((Ord C /\ B e. C) -> (A (_ B -> A e. C)))
12	11	ex 373	. . . . . . 7 |- (Ord A -> (Ord B -> ((Ord C /\ B e. C) -> (A (_ B -> A e. C))))
13		ordelord 2970	. . . . . . 7 |- ((Ord C /\ B e. C) -> Ord B)
14	12, 13	syl5 21	. . . . . 6 |- (Ord A -> ((Ord C /\ B e. C) -> ((Ord C /\ B e. C) -> (A (_ B -> A e. C))))
15	14	pm2.43d 65	. . . . 5 |- (Ord A -> ((Ord C /\ B e. C) -> (A (_ B -> A e. C)))
16	15	exp3a 375	. . . 4 |- (Ord A -> (Ord C -> (B e. C -> (A (_ B -> A e. C))))
17	16	imp 350	. . 3 |- ((Ord A /\ Ord C) -> (B e. C -> (A (_ B -> A e. C)))
18	17	com23 32	. 2 |- ((Ord A /\ Ord C) -> (A (_ B -> (B e. C -> A e. C)))
19	18	imp3a 361	1 |- ((Ord A /\ Ord C) -> ((A (_ B /\ B e. C) -> A e. C))

Syntax hints:   -> wi 3   \/ wo 222   /\ wa 223   = wceq 956   e. wcel 958   (_ wss 2047  Ord word 2947

This theorem is referenced by:  ontr2 3004  nnarcl 4232

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951

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