Theorem oaord 8555

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 8554	. . 3 ⊢ ((𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))
2	1	3adant1 1148	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 → (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))
3		oveq2 7428	. . . . . 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 8554	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ∈ 𝐴 → (𝐶 +o 𝐵) ∈ (𝐶 +o 𝐴)))
6	5	3adant2 1149	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐵 ∈ 𝐴 → (𝐶 +o 𝐵) ∈ (𝐶 +o 𝐴)))
7	4, 6	orim12d 979	. . . 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 153	. . 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 1105	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ↔ ((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On))
10		ancom 466	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) ↔ (𝐶 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)))
11		anandi 689	. . . . . 6 ⊢ ((𝐶 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) ↔ ((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐵 ∈ On)))
12	9, 10, 11	3bitri 300	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ↔ ((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐵 ∈ On)))
13		oacl 8543	. . . . . . 7 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶 +o 𝐴) ∈ On)
14		eloni 6372	. . . . . . 7 ⊢ ((𝐶 +o 𝐴) ∈ On → Ord (𝐶 +o 𝐴))
15	13, 14	syl 18	. . . . . 6 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On) → Ord (𝐶 +o 𝐴))
16		oacl 8543	. . . . . . 7 ⊢ ((𝐶 ∈ On ∧ 𝐵 ∈ On) → (𝐶 +o 𝐵) ∈ On)
17		eloni 6372	. . . . . . 7 ⊢ ((𝐶 +o 𝐵) ∈ On → Ord (𝐶 +o 𝐵))
18	16, 17	syl 18	. . . . . 6 ⊢ ((𝐶 ∈ On ∧ 𝐵 ∈ On) → Ord (𝐶 +o 𝐵))
19	15, 18	anim12i 625	. . . . 5 ⊢ (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐵 ∈ On)) → (Ord (𝐶 +o 𝐴) ∧ Ord (𝐶 +o 𝐵)))
20	12, 19	sylbi 220	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (Ord (𝐶 +o 𝐴) ∧ Ord (𝐶 +o 𝐵)))
21		ordtri2 6398	. . . 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 18	. . 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 6372	. . . . . 6 ⊢ (𝐴 ∈ On → Ord 𝐴)
24		eloni 6372	. . . . . 6 ⊢ (𝐵 ∈ On → Ord 𝐵)
25	23, 24	anim12i 625	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (Ord 𝐴 ∧ Ord 𝐵))
26	25	3adant3 1150	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (Ord 𝐴 ∧ Ord 𝐵))
27		ordtri2 6398	. . . 4 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
28	26, 27	syl 18	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
29	8, 22, 28	3imtr4d 297	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵) → 𝐴 ∈ 𝐵))
30	2, 29	impbid 215	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 ↔ (𝐶 +o 𝐴) ∈ (𝐶 +o 𝐵)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  Ord word 6361  Oncon0 6362  (class class class)co 7420   +_o coa 8473

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-oadd 8480

This theorem is used by:  oacan  8556  oaword  8557  oaord1  8559  oa00  8567  oalimcl  8568  oaass  8569  odi  8587  oneo  8589  omeulem1  8590  omeulem2  8591  oeeui  8611  omxpenlem  9097  cantnflt  9673  cantnflem1d  9689  cantnflem1  9690  oaord3  44294  oawordex2  44328