Theorem ordtr2 2257

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 2227	. . . . . . . . . 10 |- ((Ord A /\ Ord B) -> (A (_ B <-> (A e. B \/ A = B)))
2	1	biimpd 135	. . . . . . . . 9 |- ((Ord A /\ Ord B) -> (A (_ B -> (A e. B \/ A = B)))
3		ordtr1 2256	. . . . . . . . . . . . 13 |- (Ord C -> ((A e. B /\ B e. C) -> A e. C))
4	3	exp3a 292	. . . . . . . . . . . 12 |- (Ord C -> (A e. B -> (B e. C -> A e. C)))
5		eleq1a 1158	. . . . . . . . . . . . . 14 |- (B e. C -> (A = B -> A e. C))
6	5	com12 13	. . . . . . . . . . . . 13 |- (A = B -> (B e. C -> A e. C))
7	6	a1i 7	. . . . . . . . . . . 12 |- (Ord C -> (A = B -> (B e. C -> A e. C)))
8	4, 7	jaod 329	. . . . . . . . . . 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 277	. . . . . . . . 9 |- ((Ord C /\ B e. C) -> ((A e. B \/ A = B) -> A e. C))
11	2, 10	syl9 55	. . . . . . . 8 |- ((Ord A /\ Ord B) -> ((Ord C /\ B e. C) -> (A (_ B -> A e. C)))
12	11	exp 291	. . . . . . 7 |- (Ord A -> (Ord B -> ((Ord C /\ B e. C) -> (A (_ B -> A e. C))))
13		ordelord 2221	. . . . . . 7 |- ((Ord C /\ B e. C) -> Ord B)
14	12, 13	syl5 22	. . . . . 6 |- (Ord A -> ((Ord C /\ B e. C) -> ((Ord C /\ B e. C) -> (A (_ B -> A e. C))))
15	14	pm2.43d 59	. . . . 5 |- (Ord A -> ((Ord C /\ B e. C) -> (A (_ B -> A e. C)))
16	15	exp3a 292	. . . 4 |- (Ord A -> (Ord C -> (B e. C -> (A (_ B -> A e. C))))
17	16	imp 277	. . 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 279	1 |- ((Ord A /\ Ord C) -> ((A (_ B /\ B e. C) -> A e. C))

Syntax hints:   -> wi 2   \/ wo 195   /\ wa 196   = wceq 1091   e. wcel 1092   (_ wss 1487  Ord word 2198

This theorem is referenced by:  ontr2 2259  nnarcl 3174

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202

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