Theorem ordssun 4640

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 4637	. . 3 |- ( ( Ord B /\ Ord C ) -> ( B C_ C \/ C C_ B ) )
2		ssequn1 3477	. . . . . 6 |- ( B C_ C <-> ( B u. C ) = C )
3		sseq2 3330	. . . . . 6 |- ( ( B u. C ) = C -> ( A C_ ( B u. C ) <-> A C_ C ) )
4	2, 3	sylbi 188	. . . . 5 |- ( B C_ C -> ( A C_ ( B u. C ) <-> A C_ C ) )
5		olc 374	. . . . 5 |- ( A C_ C -> ( A C_ B \/ A C_ C ) )
6	4, 5	syl6bi 220	. . . 4 |- ( B C_ C -> ( A C_ ( B u. C ) -> ( A C_ B \/ A C_ C ) ) )
7		ssequn2 3480	. . . . . 6 |- ( C C_ B <-> ( B u. C ) = B )
8		sseq2 3330	. . . . . 6 |- ( ( B u. C ) = B -> ( A C_ ( B u. C ) <-> A C_ B ) )
9	7, 8	sylbi 188	. . . . 5 |- ( C C_ B -> ( A C_ ( B u. C ) <-> A C_ B ) )
10		orc 375	. . . . 5 |- ( A C_ B -> ( A C_ B \/ A C_ C ) )
11	9, 10	syl6bi 220	. . . 4 |- ( C C_ B -> ( A C_ ( B u. C ) -> ( A C_ B \/ A C_ C ) ) )
12	6, 11	jaoi 369	. . 3 |- ( ( B C_ C \/ C C_ B ) -> ( A C_ ( B u. C ) -> ( A C_ B \/ A C_ C ) ) )
13	1, 12	syl 16	. 2 |- ( ( Ord B /\ Ord C ) -> ( A C_ ( B u. C ) -> ( A C_ B \/ A C_ C ) ) )
14		ssun 3486	. 2 |- ( ( A C_ B \/ A C_ C ) -> A C_ ( B u. C ) )
15	13, 14	impbid1 195	1 |- ( ( Ord B /\ Ord C ) -> ( A C_ ( B u. C ) <-> ( A C_ B \/ A C_ C ) ) )

Syntax hints:   -> wi 4   <-> wb 177   \/ wo 358   /\ wa 359   = wceq 1649   u. cun 3278   C_ wss 3280   Ord word 4540

This theorem is referenced by:  ordsucun  4764

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-tr 4263  df-eprel 4454  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544