Theorem ordssun 3769

Description: Property of a subclass of the maximum (i.e. union) of two ordinals.

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 3768	. . 3 |- ((Ord B /\ Ord C) -> (B C_ C \/ C C_ B))
2		ssequn1 2775	. . . . . 6 |- (B C_ C <-> (B u. C) = C)
3		sseq2 2639	. . . . . 6 |- ((B u. C) = C -> (A C_ (B u. C) <-> A C_ C))
4	2, 3	sylbi 216	. . . . 5 |- (B C_ C -> (A C_ (B u. C) <-> A C_ C))
5		olc 290	. . . . 5 |- (A C_ C -> (A C_ B \/ A C_ C))
6	4, 5	syl6bi 231	. . . 4 |- (B C_ C -> (A C_ (B u. C) -> (A C_ B \/ A C_ C)))
7		ssequn2 2779	. . . . . 6 |- (C C_ B <-> (B u. C) = B)
8		sseq2 2639	. . . . . 6 |- ((B u. C) = B -> (A C_ (B u. C) <-> A C_ B))
9	7, 8	sylbi 216	. . . . 5 |- (C C_ B -> (A C_ (B u. C) <-> A C_ B))
10		orc 291	. . . . 5 |- (A C_ B -> (A C_ B \/ A C_ C))
11	9, 10	syl6bi 231	. . . 4 |- (C C_ B -> (A C_ (B u. C) -> (A C_ B \/ A C_ C)))
12	6, 11	jaoi 368	. . 3 |- ((B C_ C \/ C C_ B) -> (A C_ (B u. C) -> (A C_ B \/ A C_ C)))
13	1, 12	syl 12	. 2 |- ((Ord B /\ Ord C) -> (A C_ (B u. C) -> (A C_ B \/ A C_ C)))
14		ssun 2782	. 2 |- ((A C_ B \/ A C_ C) -> A C_ (B u. C))
15	13, 14	impbid1 575	1 |- ((Ord B /\ Ord C) -> (A C_ (B u. C) <-> (A C_ B \/ A C_ C)))

Syntax hints:   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   = wceq 1298   u. cun 2591   C_ wss 2593  Ord word 3656

This theorem is referenced by:  ordsucun 3905

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660

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