Theorem oacan 4172

Description: Left cancellation law for ordinal addition. Corollary 8.5 of [TakeutiZaring] p. 58.

Assertion

Ref	Expression
oacan	|- ((A e. On /\ B e. On /\ C e. On) -> ((A +o B) = (A +o C) <-> B = C))

Proof of Theorem oacan

Step	Hyp	Ref	Expression
1		oaord 4171	. . . . 5 |- ((B e. On /\ C e. On /\ A e. On) -> (B e. C <-> (A +o B) e. (A +o C)))
2	1	3comr 840	. . . 4 |- ((A e. On /\ B e. On /\ C e. On) -> (B e. C <-> (A +o B) e. (A +o C)))
3		oaord 4171	. . . . 5 |- ((C e. On /\ B e. On /\ A e. On) -> (C e. B <-> (A +o C) e. (A +o B)))
4	3	3com13 837	. . . 4 |- ((A e. On /\ B e. On /\ C e. On) -> (C e. B <-> (A +o C) e. (A +o B)))
5	2, 4	orbi12d 626	. . 3 |- ((A e. On /\ B e. On /\ C e. On) -> ((B e. C \/ C e. B) <-> ((A +o B) e. (A +o C) \/ (A +o C) e. (A +o B))))
6	5	negbid 610	. 2 |- ((A e. On /\ B e. On /\ C e. On) -> (-. (B e. C \/ C e. B) <-> -. ((A +o B) e. (A +o C) \/ (A +o C) e. (A +o B))))
7		ordtri3 2978	. . . 4 |- ((Ord B /\ Ord C) -> (B = C <-> -. (B e. C \/ C e. B)))
8		eloni 2953	. . . 4 |- (B e. On -> Ord B)
9		eloni 2953	. . . 4 |- (C e. On -> Ord C)
10	7, 8, 9	syl2an 454	. . 3 |- ((B e. On /\ C e. On) -> (B = C <-> -. (B e. C \/ C e. B)))
11	10	3adant1 796	. 2 |- ((A e. On /\ B e. On /\ C e. On) -> (B = C <-> -. (B e. C \/ C e. B)))
12		ordtri3 2978	. . . 4 |- ((Ord (A +o B) /\ Ord (A +o C)) -> ((A +o B) = (A +o C) <-> -. ((A +o B) e. (A +o C) \/ (A +o C) e. (A +o B))))
13		oacl 4160	. . . . 5 |- ((A e. On /\ B e. On) -> (A +o B) e. On)
14		eloni 2953	. . . . 5 |- ((A +o B) e. On -> Ord (A +o B))
15	13, 14	syl 10	. . . 4 |- ((A e. On /\ B e. On) -> Ord (A +o B))
16		oacl 4160	. . . . 5 |- ((A e. On /\ C e. On) -> (A +o C) e. On)
17		eloni 2953	. . . . 5 |- ((A +o C) e. On -> Ord (A +o C))
18	16, 17	syl 10	. . . 4 |- ((A e. On /\ C e. On) -> Ord (A +o C))
19	12, 15, 18	syl2an 454	. . 3 |- (((A e. On /\ B e. On) /\ (A e. On /\ C e. On)) -> ((A +o B) = (A +o C) <-> -. ((A +o B) e. (A +o C) \/ (A +o C) e. (A +o B))))
20	19	3impdi 878	. 2 |- ((A e. On /\ B e. On /\ C e. On) -> ((A +o B) = (A +o C) <-> -. ((A +o B) e. (A +o C) \/ (A +o C) e. (A +o B))))
21	6, 11, 20	3bitr4rd 550	1 |- ((A e. On /\ B e. On /\ C e. On) -> ((A +o B) = (A +o C) <-> B = C))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   /\ w3a 774   = wceq 954   e. wcel 956  Ord word 2942  Oncon0 2943  (class class class)co 3954   +o coa 4120

This theorem is referenced by:  oaword 4173  oawordeulem 4178  nnacan 4232

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-9 963  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-rep 2688  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861

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-rab 1649  df-v 1808  df-sbc 1938  df-csb 1998  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-iun 2563  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-id 2830  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-fv 3193  df-rdg 3923  df-opr 3956  df-oprab 3957  df-oadd 4125

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