Theorem oaord 3149

Description: Ordering property of ordinal addition. Proposition 8.4 of [TakeutiZaring] p. 58 and its converse.

Assertion

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

Proof of Theorem oaord

Step	Hyp	Ref	Expression
1		oaordi 3148	. . 3 ⊢ ((B ∈ On ∧ C ∈ On) → (A ∈ B → (C +o A) ∈ (C +o B)))
2	1	3adant1 597	. 2 ⊢ ((A ∈ On ∧ B ∈ On ∧ C ∈ On) → (A ∈ B → (C +o A) ∈ (C +o B)))
3		opreq2 3007	. . . . . 6 ⊢ (A = B → (C +o A) = (C +o B))
4	3	a1i 7	. . . . 5 ⊢ ((A ∈ On ∧ B ∈ On ∧ C ∈ On) → (A = B → (C +o A) = (C +o B)))
5		oaordi 3148	. . . . . 6 ⊢ ((A ∈ On ∧ C ∈ On) → (B ∈ A → (C +o B) ∈ (C +o A)))
6	5	3adant2 598	. . . . 5 ⊢ ((A ∈ On ∧ B ∈ On ∧ C ∈ On) → (B ∈ A → (C +o B) ∈ (C +o A)))
7	4, 6	orim12d 436	. . . 4 ⊢ ((A ∈ On ∧ B ∈ On ∧ C ∈ On) → ((A = B ∨ B ∈ A) → ((C +o A) = (C +o B) ∨ (C +o B) ∈ (C +o A))))
8	7	con3d 87	. . 3 ⊢ ((A ∈ On ∧ B ∈ On ∧ C ∈ On) → (¬ ((C +o A) = (C +o B) ∨ (C +o B) ∈ (C +o A)) → ¬ (A = B ∨ B ∈ A)))
9		df-3an 583	. . . . . 6 ⊢ ((A ∈ On ∧ B ∈ On ∧ C ∈ On) ↔ ((A ∈ On ∧ B ∈ On) ∧ C ∈ On))
10		ancom 333	. . . . . 6 ⊢ (((A ∈ On ∧ B ∈ On) ∧ C ∈ On) ↔ (C ∈ On ∧ (A ∈ On ∧ B ∈ On)))
11		anandi 392	. . . . . 6 ⊢ ((C ∈ On ∧ (A ∈ On ∧ B ∈ On)) ↔ ((C ∈ On ∧ A ∈ On) ∧ (C ∈ On ∧ B ∈ On)))
12	9, 10, 11	3bitr 155	. . . . 5 ⊢ ((A ∈ On ∧ B ∈ On ∧ C ∈ On) ↔ ((C ∈ On ∧ A ∈ On) ∧ (C ∈ On ∧ B ∈ On)))
13		oacl 3138	. . . . . . 7 ⊢ ((C ∈ On ∧ A ∈ On) → (C +o A) ∈ On)
14		eloni 2209	. . . . . . 7 ⊢ ((C +o A) ∈ On → Ord (C +o A))
15	13, 14	syl 12	. . . . . 6 ⊢ ((C ∈ On ∧ A ∈ On) → Ord (C +o A))
16		oacl 3138	. . . . . . 7 ⊢ ((C ∈ On ∧ B ∈ On) → (C +o B) ∈ On)
17		eloni 2209	. . . . . . 7 ⊢ ((C +o B) ∈ On → Ord (C +o B))
18	16, 17	syl 12	. . . . . 6 ⊢ ((C ∈ On ∧ B ∈ On) → Ord (C +o B))
19	15, 18	anim12i 268	. . . . 5 ⊢ (((C ∈ On ∧ A ∈ On) ∧ (C ∈ On ∧ B ∈ On)) → (Ord (C +o A) ∧ Ord (C +o B)))
20	12, 19	sylbi 174	. . . 4 ⊢ ((A ∈ On ∧ B ∈ On ∧ C ∈ On) → (Ord (C +o A) ∧ Ord (C +o B)))
21		ordtri2 2233	. . . 4 ⊢ ((Ord (C +o A) ∧ Ord (C +o B)) → ((C +o A) ∈ (C +o B) ↔ ¬ ((C +o A) = (C +o B) ∨ (C +o B) ∈ (C +o A))))
22	20, 21	syl 12	. . 3 ⊢ ((A ∈ On ∧ B ∈ On ∧ C ∈ On) → ((C +o A) ∈ (C +o B) ↔ ¬ ((C +o A) = (C +o B) ∨ (C +o B) ∈ (C +o A))))
23		3simpa 591	. . . 4 ⊢ ((A ∈ On ∧ B ∈ On ∧ C ∈ On) → (A ∈ On ∧ B ∈ On))
24		eloni 2209	. . . . 5 ⊢ (A ∈ On → Ord A)
25		eloni 2209	. . . . 5 ⊢ (B ∈ On → Ord B)
26	24, 25	anim12i 268	. . . 4 ⊢ ((A ∈ On ∧ B ∈ On) → (Ord A ∧ Ord B))
27		ordtri2 2233	. . . 4 ⊢ ((Ord A ∧ Ord B) → (A ∈ B ↔ ¬ (A = B ∨ B ∈ A)))
28	23, 26, 27	3syl 21	. . 3 ⊢ ((A ∈ On ∧ B ∈ On ∧ C ∈ On) → (A ∈ B ↔ ¬ (A = B ∨ B ∈ A)))
29	8, 22, 28	3imtr4d 421	. 2 ⊢ ((A ∈ On ∧ B ∈ On ∧ C ∈ On) → ((C +o A) ∈ (C +o B) → A ∈ B))
30	2, 29	impbid 397	1 ⊢ ((A ∈ On ∧ B ∈ On ∧ C ∈ On) → (A ∈ B ↔ (C +o A) ∈ (C +o B)))

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:  oacan 3150  oaword 3151  oaord1 3153  oa00 3161  oalimcl 3162  oaass 3163  nnaord 3177

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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