Theorem ordequn 3086

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 3084	. 2 |- ((Ord B /\ Ord C) -> (B (_ C \/ C (_ B))
2		ssequn1 2203	. . . . 5 |- (B (_ C <-> (B u. C) = C)
3		eqeq2 1487	. . . . 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 2206	. . . . 5 |- (C (_ B <-> (B u. C) = B)
8		eqeq2 1487	. . . . 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 958   u. cun 2048   (_ wss 2050  Ord word 2953

This theorem is referenced by:  ordun 3087

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957

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