Theorem omord2 8508

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 8507	. . 3 ⊢ (((𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ 𝐵 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))
2	1	3adantl1 1167	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ 𝐵 → (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))
3		oveq2 7377	. . . . . 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 8507	. . . . . 6 ⊢ (((𝐴 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐵 ∈ 𝐴 → (𝐶 ·o 𝐵) ∈ (𝐶 ·o 𝐴)))
6	5	3adantl2 1168	. . . . 5 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐵 ∈ 𝐴 → (𝐶 ·o 𝐵) ∈ (𝐶 ·o 𝐴)))
7	4, 6	orim12d 966	. . . 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 152	. . 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 8477	. . . . . . . 8 ⊢ ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶 ·o 𝐴) ∈ On)
10		omcl 8477	. . . . . . . 8 ⊢ ((𝐶 ∈ On ∧ 𝐵 ∈ On) → (𝐶 ·o 𝐵) ∈ On)
11		eloni 6330	. . . . . . . . 9 ⊢ ((𝐶 ·o 𝐴) ∈ On → Ord (𝐶 ·o 𝐴))
12		eloni 6330	. . . . . . . . 9 ⊢ ((𝐶 ·o 𝐵) ∈ On → Ord (𝐶 ·o 𝐵))
13		ordtri2 6355	. . . . . . . . 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 596	. . . . . . . 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 596	. . . . . . 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 678	. . . . . 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 458	. . . . 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 1109	. . . 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 480	. . 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 6330	. . . . . 6 ⊢ (𝐴 ∈ On → Ord 𝐴)
21		eloni 6330	. . . . . 6 ⊢ (𝐵 ∈ On → Ord 𝐵)
22		ordtri2 6355	. . . . . 6 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
23	20, 21, 22	syl2an 596	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
24	23	3adant3 1132	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴 ∈ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
25	24	adantr 480	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴)))
26	8, 19, 25	3imtr4d 294	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → ((𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵) → 𝐴 ∈ 𝐵))
27	2, 26	impbid 212	1 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 ∈ 𝐵 ↔ (𝐶 ·o 𝐴) ∈ (𝐶 ·o 𝐵)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∅c0 4292  Ord word 6319  Oncon0 6320  (class class class)co 7369   ·_o comu 8409

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-om 7823  df-2nd 7948  df-frecs 8237  df-wrecs 8268  df-recs 8317  df-rdg 8355  df-oadd 8415  df-omul 8416

This theorem is referenced by:  omord  8509  omword  8511  oeeui  8543  omabs  8592  omxpenlem  9019  cantnflt  9601  cnfcom  9629