Theorem ordssun 3079

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 3078	. . 3 |- ((Ord B /\ Ord C) -> (B (_ C \/ C (_ B))
2		ssequn1 2200	. . . . . 6 |- (B (_ C <-> (B u. C) = C)
3		sseq2 2083	. . . . . 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 2203	. . . . . 6 |- (C (_ B <-> (B u. C) = B)
8		sseq2 2083	. . . . . 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 2206	. 2 |- ((A (_ B \/ A (_ C) -> A (_ (B u. C))
15	13, 14	impbid1 517	1 |- ((Ord B /\ Ord C) -> (A (_ (B u. C) <-> (A (_ B \/ A (_ C)))

Syntax hints:   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   = wceq 956   u. cun 2045   (_ wss 2047  Ord word 2947

This theorem is referenced by:  ordsucun 3082

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951

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