Theorem omord2 8557

Description: Ordering property of ordinal multiplication. (Contributed by NM, 25-Dec-2004.)

Assertion

Ref	Expression
omord2	⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ 𝐵 ↔ (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))

Proof of Theorem omord2

Step	Hyp	Ref	Expression
1		omordi 8556	. . 3 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ 𝐵 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))
2	1	3adantl1 1185	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ 𝐵 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))
3		oveq2 7422	. . . . . 6 ⊢ (𝐴 = 𝐵 → (𝐶 ·o 𝐴) = (𝐶 ·o 𝐵))
4	3	a1i 11	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 = 𝐵 → (𝐶 ·o 𝐴) = (𝐶 ·o 𝐵)))
5		omordi 8556	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐵 ∈ 𝐴 → (𝐶 ·o 𝐵) ∈ (𝐶 ·o 𝐴)))
6	5	3adantl2 1186	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐵 ∈ 𝐴 → (𝐶 ·o 𝐵) ∈ (𝐶 ·o 𝐴)))
7	4, 6	orim12d 979	. . . 4 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → ((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) → (¬ ((𝐶 ·o 𝐴) = (𝐶 ·o 𝐵) ∨ (𝐶 ·o 𝐵) ∈ (𝐶 ·o 𝐴)) → ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
9		omcl 8526	. . . . . . . 8 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶 ·o 𝐴) ∈ On)
10		omcl 8526	. . . . . . . 8 ⊢ ((𝐶 ∈ On ∧ 𝐵 ∈ On) → (𝐶 ·o 𝐵) ∈ On)
11		eloni 6367	. . . . . . . . 9 ⊢ ((𝐶 ·o 𝐴) ∈ On → Ord (𝐶 ·o 𝐴))
12		eloni 6367	. . . . . . . . 9 ⊢ ((𝐶 ·o 𝐵) ∈ On → Ord (𝐶 ·o 𝐵))
13		ordtri2 6393	. . . . . . . . 9 ⊢ ((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))))
14	11, 12, 13	syl2an 608	. . . . . . . 8 ⊢ (((𝐶 ·o 𝐴) ∈ On ∧ (𝐶 ·o 𝐵) ∈ On) → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) ↔ ¬ ((𝐶 ·o 𝐴) = (𝐶 ·o 𝐵) ∨ (𝐶 ·o 𝐵) ∈ (𝐶 ·o 𝐴))))
15	9, 10, 14	syl2an 608	. . . . . . 7 ⊢ (((𝐶 ∈ On ∧ 𝐴 ∈ On) ∧ (𝐶 ∈ On ∧ 𝐵 ∈ On)) → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) ↔ ¬ ((𝐶 ·o 𝐴) = (𝐶 ·o 𝐵) ∨ (𝐶 ·o 𝐵) ∈ (𝐶 ·o 𝐴))))
16	15	anandis 691	. . . . . 6 ⊢ ((𝐶 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) ↔ ¬ ((𝐶 ·o 𝐴) = (𝐶 ·o 𝐵) ∨ (𝐶 ·o 𝐵) ∈ (𝐶 ·o 𝐴))))
17	16	ancoms 464	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) ↔ ¬ ((𝐶 ·o 𝐴) = (𝐶 ·o 𝐵) ∨ (𝐶 ·o 𝐵) ∈ (𝐶 ·o 𝐴))))
18	17	3impa 1127	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) ↔ ¬ ((𝐶 ·o 𝐴) = (𝐶 ·o 𝐵) ∨ (𝐶 ·o 𝐵) ∈ (𝐶 ·o 𝐴))))
19	18	adantr 486	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) ↔ ¬ ((𝐶 ·o 𝐴) = (𝐶 ·o 𝐵) ∨ (𝐶 ·o 𝐵) ∈ (𝐶 ·o 𝐴))))
20		eloni 6367	. . . . . 6 ⊢ (𝐴 ∈ On → Ord 𝐴)
21		eloni 6367	. . . . . 6 ⊢ (𝐵 ∈ On → Ord 𝐵)
22		ordtri2 6393	. . . . . 6 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
23	20, 21, 22	syl2an 608	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
24	23	3adant3 1150	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
25	24	adantr 486	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
26	8, 19, 25	3imtr4d 297	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) → 𝐴 ∈ 𝐵))
27	2, 26	impbid 215	1 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (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 1570   ∈ wcel 2145  ∅c0 4279  Ord word 6356  Oncon0 6357  (class class class)co 7414   ·_o comu 8456

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  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 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7864  df-2nd 7988  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-oadd 8462  df-omul 8463

This theorem is used by:  omord  8558  omword  8560  oeeui  8593  omabs  8642  omxpenlem  9079  cantnflt  9654  cnfcom  9682