Theorem ordssun 3075

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

Assertion

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

Proof of Theorem ordssun

Step	Hyp	Ref	Expression
1		ordtri2or2 3074	. . 3 |- ((Ord B /\ Ord C) -> (B (_ C \/ C (_ B))
2		ssequn1 2197	. . . . . 6 |- (B (_ C <-> (B u. C) = C)
3		sseq2 2080	. . . . . 6 |- ((B u. C) = C -> (A (_ (B u. C) <-> A (_ C))
4	2, 3	sylbi 199	. . . . 5 |- (B (_ C -> (A (_ (B u. C) <-> A (_ C))
5		olc 268	. . . . 5 |- (A (_ C -> (A (_ B \/ A (_ C))
6	4, 5	syl6bi 214	. . . 4 |- (B (_ C -> (A (_ (B u. C) -> (A (_ B \/ A (_ C)))
7		ssequn2 2200	. . . . . 6 |- (C (_ B <-> (B u. C) = B)
8		sseq2 2080	. . . . . 6 |- ((B u. C) = B -> (A (_ (B u. C) <-> A (_ B))
9	7, 8	sylbi 199	. . . . 5 |- (C (_ B -> (A (_ (B u. C) <-> A (_ B))
10		orc 269	. . . . 5 |- (A (_ B -> (A (_ B \/ A (_ C))
11	9, 10	syl6bi 214	. . . 4 |- (C (_ B -> (A (_ (B u. C) -> (A (_ B \/ A (_ C)))
12	6, 11	jaoi 341	. . 3 |- ((B (_ C \/ C (_ B) -> (A (_ (B u. C) -> (A (_ B \/ A (_ C)))
13	1, 12	syl 10	. 2 |- ((Ord B /\ Ord C) -> (A (_ (B u. C) -> (A (_ B \/ A (_ C)))
14		ssun 2203	. 2 |- ((A (_ B \/ A (_ C) -> A (_ (B u. C))
15	13, 14	impbid1 516	1 |- ((Ord B /\ Ord C) -> (A (_ (B u. C) <-> (A (_ B \/ A (_ C)))

Syntax hints:   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   = wceq 955   u. cun 2042   (_ wss 2044  Ord word 2943

This theorem is referenced by:  ordsucun 3078

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-pr 2775

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-tr 2677  df-eprel 2828  df-po 2836  df-so 2846  df-fr 2913  df-we 2930  df-ord 2947

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