Theorem ordequn 3075

Description: The maximum (i.e. union) of two ordinals is either one or the other. Similar to Exercise 14 of [TakeutiZaring] p. 40.

Assertion

Ref	Expression
ordequn	|- ((Ord B /\ Ord C) -> (A = (B u. C) -> (A = B \/ A = C)))

Proof of Theorem ordequn

Step	Hyp	Ref	Expression
1		ordtri2or2 3073	. 2 |- ((Ord B /\ Ord C) -> (B (_ C \/ C (_ B))
2		ssequn1 2196	. . . . 5 |- (B (_ C <-> (B u. C) = C)
3		eqeq2 1481	. . . . 5 |- ((B u. C) = C -> (A = (B u. C) <-> A = C))
4	2, 3	sylbi 199	. . . 4 |- (B (_ C -> (A = (B u. C) <-> A = C))
5		olc 268	. . . 4 |- (A = C -> (A = B \/ A = C))
6	4, 5	syl6bi 214	. . 3 |- (B (_ C -> (A = (B u. C) -> (A = B \/ A = C)))
7		ssequn2 2199	. . . . 5 |- (C (_ B <-> (B u. C) = B)
8		eqeq2 1481	. . . . 5 |- ((B u. C) = B -> (A = (B u. C) <-> A = B))
9	7, 8	sylbi 199	. . . 4 |- (C (_ B -> (A = (B u. C) <-> A = B))
10		orc 269	. . . 4 |- (A = B -> (A = B \/ A = C))
11	9, 10	syl6bi 214	. . 3 |- (C (_ B -> (A = (B u. C) -> (A = B \/ A = C)))
12	6, 11	jaoi 341	. 2 |- ((B (_ C \/ C (_ B) -> (A = (B u. C) -> (A = B \/ A = C)))
13	1, 12	syl 10	1 |- ((Ord B /\ Ord C) -> (A = (B u. C) -> (A = B \/ A = C)))

Syntax hints:   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   = wceq 954   u. cun 2041   (_ wss 2043  Ord word 2942

This theorem is referenced by:  ordun 3076

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946

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