Theorem oaord 8472

Description: Ordering property of ordinal addition. Proposition 8.4 of [TakeutiZaring] p. 58 and its converse. (Contributed by NM, 5-Dec-2004.)

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 8471	. . 3 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))
2	1	3adant1 1130	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))
3		oveq2 7364	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐶 +o 𝐴) = (𝐶 +o 𝐵))
4	3	a1i 11	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 = 𝐵 → (𝐶 +o 𝐴) = (𝐶 +o 𝐵)))
5		oaordi 8471	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ∈ 𝐴 → (𝐶 +o 𝐵) ∈ (𝐶 +o 𝐴)))
6	5	3adant2 1131	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ∈ 𝐴 → (𝐶 +o 𝐵) ∈ (𝐶 +o 𝐴)))
7	4, 6	orim12d 966	. . . 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 152	. . 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 1088	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ↔ ((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On))
10		ancom 460	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) ↔ (𝐶 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)))
11		anandi 676	. . . . . 6 ⊢ ((𝐶 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ↔ ((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐵 ∈ On)))
12	9, 10, 11	3bitri 297	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ↔ ((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐵 ∈ On)))
13		oacl 8460	. . . . . . 7 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶 +o 𝐴) ∈ On)
14		eloni 6325	. . . . . . 7 ⊢ ((𝐶 +o 𝐴) ∈ On → Ord (𝐶 +o 𝐴))
15	13, 14	syl 17	. . . . . 6 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On) → Ord (𝐶 +o 𝐴))
16		oacl 8460	. . . . . . 7 ⊢ ((𝐶 ∈ On ∧ 𝐵 ∈ On) → (𝐶 +o 𝐵) ∈ On)
17		eloni 6325	. . . . . . 7 ⊢ ((𝐶 +o 𝐵) ∈ On → Ord (𝐶 +o 𝐵))
18	16, 17	syl 17	. . . . . 6 ⊢ ((𝐶 ∈ On ∧ 𝐵 ∈ On) → Ord (𝐶 +o 𝐵))
19	15, 18	anim12i 613	. . . . 5 ⊢ (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐵 ∈ On)) → (Ord (𝐶 +o 𝐴) ∧ Ord (𝐶 +o 𝐵)))
20	12, 19	sylbi 217	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (Ord (𝐶 +o 𝐴) ∧ Ord (𝐶 +o 𝐵)))
21		ordtri2 6350	. . . 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 17	. . 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		eloni 6325	. . . . . 6 ⊢ (𝐴 ∈ On → Ord 𝐴)
24		eloni 6325	. . . . . 6 ⊢ (𝐵 ∈ On → Ord 𝐵)
25	23, 24	anim12i 613	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (Ord 𝐴 ∧ Ord 𝐵))
26	25	3adant3 1132	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (Ord 𝐴 ∧ Ord 𝐵))
27		ordtri2 6350	. . . 4 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
28	26, 27	syl 17	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
29	8, 22, 28	3imtr4d 294	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵) → 𝐴 ∈ 𝐵))
30	2, 29	impbid 212	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 ↔ (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  Ord word 6314  Oncon0 6315  (class class class)co 7356   +_o coa 8392

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-oadd 8399

This theorem is referenced by:  oacan  8473  oaword  8474  oaord1  8476  oa00  8484  oalimcl  8485  oaass  8486  odi  8504  oneo  8506  omeulem1  8507  omeulem2  8508  oeeui  8528  omxpenlem  9004  cantnflt  9579  cantnflem1d  9595  cantnflem1  9596  oaord3  43476  oawordex2  43510