Theorem oacan 4182

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 4181	. . . . 5 |- ((B e. On /\ C e. On /\ A e. On) -> (B e. C <-> (A +o B) e. (A +o C)))
2	1	3comr 841	. . . 4 |- ((A e. On /\ B e. On /\ C e. On) -> (B e. C <-> (A +o B) e. (A +o C)))
3		oaord 4181	. . . . 5 |- ((C e. On /\ B e. On /\ A e. On) -> (C e. B <-> (A +o C) e. (A +o B)))
4	3	3com13 838	. . . 4 |- ((A e. On /\ B e. On /\ C e. On) -> (C e. B <-> (A +o C) e. (A +o B)))
5	2, 4	orbi12d 627	. . 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 611	. 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 2983	. . . 4 |- ((Ord B /\ Ord C) -> (B = C <-> -. (B e. C \/ C e. B)))
8		eloni 2958	. . . 4 |- (B e. On -> Ord B)
9		eloni 2958	. . . 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 797	. 2 |- ((A e. On /\ B e. On /\ C e. On) -> (B = C <-> -. (B e. C \/ C e. B)))
12		ordtri3 2983	. . . 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 4170	. . . . 5 |- ((A e. On /\ B e. On) -> (A +o B) e. On)
14		eloni 2958	. . . . 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 4170	. . . . 5 |- ((A e. On /\ C e. On) -> (A +o C) e. On)
17		eloni 2958	. . . . 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 880	. 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 551	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 775   = wceq 956   e. wcel 958  Ord word 2947  Oncon0 2948  (class class class)co 3963   +o coa 4130

This theorem is referenced by:  oaword 4183  oawordeulem 4188  nnacan 4242

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-9 965  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-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

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-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198  df-rdg 3932  df-opr 3965  df-oprab 3966  df-oadd 4135

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