Theorem ordssun 5522

Description: Property of a subclass of the maximum (i.e. union) of two ordinals. (Contributed by NM, 28-Nov-2003.)

Assertion

Ref	Expression
ordssun	|- ( ( Ord B /\ Ord C ) -> ( A C_ ( B u. C ) <-> ( A C_ B \/ A C_ C ) ) )

Proof of Theorem ordssun

Step	Hyp	Ref	Expression
1		ordtri2or2 5519	. . 3 |- ( ( Ord B /\ Ord C ) -> ( B C_ C \/ C C_ B ) )
2		ssequn1 3604	. . . . . 6 |- ( B C_ C <-> ( B u. C ) = C )
3		sseq2 3454	. . . . . 6 |- ( ( B u. C ) = C -> ( A C_ ( B u. C ) <-> A C_ C ) )
4	2, 3	sylbi 199	. . . . 5 |- ( B C_ C -> ( A C_ ( B u. C ) <-> A C_ C ) )
5		olc 386	. . . . 5 |- ( A C_ C -> ( A C_ B \/ A C_ C ) )
6	4, 5	syl6bi 232	. . . 4 |- ( B C_ C -> ( A C_ ( B u. C ) -> ( A C_ B \/ A C_ C ) ) )
7		ssequn2 3607	. . . . . 6 |- ( C C_ B <-> ( B u. C ) = B )
8		sseq2 3454	. . . . . 6 |- ( ( B u. C ) = B -> ( A C_ ( B u. C ) <-> A C_ B ) )
9	7, 8	sylbi 199	. . . . 5 |- ( C C_ B -> ( A C_ ( B u. C ) <-> A C_ B ) )
10		orc 387	. . . . 5 |- ( A C_ B -> ( A C_ B \/ A C_ C ) )
11	9, 10	syl6bi 232	. . . 4 |- ( C C_ B -> ( A C_ ( B u. C ) -> ( A C_ B \/ A C_ C ) ) )
12	6, 11	jaoi 381	. . 3 |- ( ( B C_ C \/ C C_ B ) -> ( A C_ ( B u. C ) -> ( A C_ B \/ A C_ C ) ) )
13	1, 12	syl 17	. 2 |- ( ( Ord B /\ Ord C ) -> ( A C_ ( B u. C ) -> ( A C_ B \/ A C_ C ) ) )
14		ssun 3613	. 2 |- ( ( A C_ B \/ A C_ C ) -> A C_ ( B u. C ) )
15	13, 14	impbid1 207	1 |- ( ( Ord B /\ Ord C ) -> ( A C_ ( B u. C ) <-> ( A C_ B \/ A C_ C ) ) )

Syntax hints:   -> wi 4   <-> wb 188   \/ wo 370   /\ wa 371   = wceq 1444   u. cun 3402   C_ wss 3404   Ord word 5422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-tr 4498  df-eprel 4745  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-ord 5426

This theorem is referenced by:  ordsucun  6652