Theorem oacan 3150

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

Assertion

Ref	Expression
oacan	⊢ ((A ∈ On ∧ B ∈ On ∧ C ∈ On) → ((A +o B) = (A +o C) ↔ B = C))

Proof of Theorem oacan

Step	Hyp	Ref	Expression
1		oaord 3149	. . . . . 6 ⊢ ((B ∈ On ∧ C ∈ On ∧ A ∈ On) → (B ∈ C ↔ (A +o B) ∈ (A +o C)))
2	1	3comr 618	. . . . 5 ⊢ ((A ∈ On ∧ B ∈ On ∧ C ∈ On) → (B ∈ C ↔ (A +o B) ∈ (A +o C)))
3		oaord 3149	. . . . . 6 ⊢ ((C ∈ On ∧ B ∈ On ∧ A ∈ On) → (C ∈ B ↔ (A +o C) ∈ (A +o B)))
4	3	3com13 615	. . . . 5 ⊢ ((A ∈ On ∧ B ∈ On ∧ C ∈ On) → (C ∈ B ↔ (A +o C) ∈ (A +o B)))
5	2, 4	orbi12d 475	. . . 4 ⊢ ((A ∈ On ∧ B ∈ On ∧ C ∈ On) → ((B ∈ C ∨ C ∈ B) ↔ ((A +o B) ∈ (A +o C) ∨ (A +o C) ∈ (A +o B))))
6	5	negbid 463	. . 3 ⊢ ((A ∈ On ∧ B ∈ On ∧ C ∈ On) → (¬ (B ∈ C ∨ C ∈ B) ↔ ¬ ((A +o B) ∈ (A +o C) ∨ (A +o C) ∈ (A +o B))))
7	6	bicomd 399	. 2 ⊢ ((A ∈ On ∧ B ∈ On ∧ C ∈ On) → (¬ ((A +o B) ∈ (A +o C) ∨ (A +o C) ∈ (A +o B)) ↔ ¬ (B ∈ C ∨ C ∈ B)))
8		ordtri3 2234	. . . 4 ⊢ ((Ord (A +o B) ∧ Ord (A +o C)) → ((A +o B) = (A +o C) ↔ ¬ ((A +o B) ∈ (A +o C) ∨ (A +o C) ∈ (A +o B))))
9		oacl 3138	. . . . 5 ⊢ ((A ∈ On ∧ B ∈ On) → (A +o B) ∈ On)
10		eloni 2209	. . . . 5 ⊢ ((A +o B) ∈ On → Ord (A +o B))
11	9, 10	syl 12	. . . 4 ⊢ ((A ∈ On ∧ B ∈ On) → Ord (A +o B))
12		oacl 3138	. . . . 5 ⊢ ((A ∈ On ∧ C ∈ On) → (A +o C) ∈ On)
13		eloni 2209	. . . . 5 ⊢ ((A +o C) ∈ On → Ord (A +o C))
14	12, 13	syl 12	. . . 4 ⊢ ((A ∈ On ∧ C ∈ On) → Ord (A +o C))
15	8, 11, 14	syl2an 349	. . 3 ⊢ (((A ∈ On ∧ B ∈ On) ∧ (A ∈ On ∧ C ∈ On)) → ((A +o B) = (A +o C) ↔ ¬ ((A +o B) ∈ (A +o C) ∨ (A +o C) ∈ (A +o B))))
16	15	3impdi 630	. 2 ⊢ ((A ∈ On ∧ B ∈ On ∧ C ∈ On) → ((A +o B) = (A +o C) ↔ ¬ ((A +o B) ∈ (A +o C) ∨ (A +o C) ∈ (A +o B))))
17		ordtri3 2234	. . . 4 ⊢ ((Ord B ∧ Ord C) → (B = C ↔ ¬ (B ∈ C ∨ C ∈ B)))
18		eloni 2209	. . . 4 ⊢ (B ∈ On → Ord B)
19		eloni 2209	. . . 4 ⊢ (C ∈ On → Ord C)
20	17, 18, 19	syl2an 349	. . 3 ⊢ ((B ∈ On ∧ C ∈ On) → (B = C ↔ ¬ (B ∈ C ∨ C ∈ B)))
21	20	3adant1 597	. 2 ⊢ ((A ∈ On ∧ B ∈ On ∧ C ∈ On) → (B = C ↔ ¬ (B ∈ C ∨ C ∈ B)))
22	7, 16, 21	3bitr4d 424	1 ⊢ ((A ∈ On ∧ B ∈ On ∧ C ∈ On) → ((A +o B) = (A +o C) ↔ B = C))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∨ wo 195   ∧ wa 196   ∧ w3a 581   = wceq 1091   ∈ wcel 1092  Ord word 2198  Oncon0 2199  (class class class)co 3001   +_o coa 3101

This theorem is referenced by:  oaword 3151  oawordeulem 3156  nnacan 3184

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-iun 1996  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-fv 2438  df-rdg 2970  df-opr 3003  df-oprab 3004  df-oadd 3106

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