Theorem ordtr2 5768

Description: Transitive law for ordinal classes. (Contributed by NM, 12-Dec-2004.) (Proof 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 5745	. . . . . . . 8 |- ( ( Ord C /\ B e. C ) -> Ord B )
2	1	ex 450	. . . . . . 7 |- ( Ord C -> ( B e. C -> Ord B ) )
3	2	ancld 576	. . . . . 6 |- ( Ord C -> ( B e. C -> ( B e. C /\ Ord B ) ) )
4	3	anc2li 580	. . . . 5 |- ( Ord C -> ( B e. C -> ( Ord C /\ ( B e. C /\ Ord B ) ) ) )
5		ordelpss 5751	. . . . . . . . . 10 |- ( ( Ord B /\ Ord C ) -> ( B e. C <-> B C. C ) )
6		sspsstr 3712	. . . . . . . . . . 11 |- ( ( A C_ B /\ B C. C ) -> A C. C )
7	6	expcom 451	. . . . . . . . . 10 |- ( B C. C -> ( A C_ B -> A C. C ) )
8	5, 7	syl6bi 243	. . . . . . . . 9 |- ( ( Ord B /\ Ord C ) -> ( B e. C -> ( A C_ B -> A C. C ) ) )
9	8	expcom 451	. . . . . . . 8 |- ( Ord C -> ( Ord B -> ( B e. C -> ( A C_ B -> A C. C ) ) ) )
10	9	com23 86	. . . . . . 7 |- ( Ord C -> ( B e. C -> ( Ord B -> ( A C_ B -> A C. C ) ) ) )
11	10	imp32 449	. . . . . 6 |- ( ( Ord C /\ ( B e. C /\ Ord B ) ) -> ( A C_ B -> A C. C ) )
12	11	com12 32	. . . . 5 |- ( A C_ B -> ( ( Ord C /\ ( B e. C /\ Ord B ) ) -> A C. C ) )
13	4, 12	syl9 77	. . . 4 |- ( Ord C -> ( A C_ B -> ( B e. C -> A C. C ) ) )
14	13	impd 447	. . 3 |- ( Ord C -> ( ( A C_ B /\ B e. C ) -> A C. C ) )
15	14	adantl 482	. 2 |- ( ( Ord A /\ Ord C ) -> ( ( A C_ B /\ B e. C ) -> A C. C ) )
16		ordelpss 5751	. 2 |- ( ( Ord A /\ Ord C ) -> ( A e. C <-> A C. C ) )
17	15, 16	sylibrd 249	1 |- ( ( Ord A /\ Ord C ) -> ( ( A C_ B /\ B e. C ) -> A e. C ) )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   C_ wss 3574   C. wpss 3575   Ord word 5722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726

This theorem is referenced by:  ordtr3OLD  5770  ontr2  5772  ordelinel  5825  ordelinelOLD  5826  smogt  7464  smorndom  7465  nnarcl  7696  nnawordex  7717  coftr  9095  noetalem3  31865  hfuni  32291