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Theorem omword 7695
Description: Weak ordering property of ordinal multiplication. (Contributed by NM, 21-Dec-2004.)
Assertion
Ref Expression
omword (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 ↔ (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵)))

Proof of Theorem omword
StepHypRef Expression
1 omord2 7692 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 ↔ (𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵)))
2 3anrot 1060 . . . . 5 ((𝐶 ∈ On ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On) ↔ (𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On))
3 omcan 7694 . . . . 5 (((𝐶 ∈ On ∧ 𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐶) → ((𝐶 ·𝑜 𝐴) = (𝐶 ·𝑜 𝐵) ↔ 𝐴 = 𝐵))
42, 3sylanbr 489 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → ((𝐶 ·𝑜 𝐴) = (𝐶 ·𝑜 𝐵) ↔ 𝐴 = 𝐵))
54bicomd 213 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴 = 𝐵 ↔ (𝐶 ·𝑜 𝐴) = (𝐶 ·𝑜 𝐵)))
61, 5orbi12d 746 . 2 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → ((𝐴𝐵𝐴 = 𝐵) ↔ ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) ∨ (𝐶 ·𝑜 𝐴) = (𝐶 ·𝑜 𝐵))))
7 onsseleq 5803 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
873adant3 1101 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
98adantr 480 . 2 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 ↔ (𝐴𝐵𝐴 = 𝐵)))
10 omcl 7661 . . . . . . 7 ((𝐶 ∈ On ∧ 𝐴 ∈ On) → (𝐶 ·𝑜 𝐴) ∈ On)
11 omcl 7661 . . . . . . 7 ((𝐶 ∈ On ∧ 𝐵 ∈ On) → (𝐶 ·𝑜 𝐵) ∈ On)
1210, 11anim12dan 900 . . . . . 6 ((𝐶 ∈ On ∧ (𝐴 ∈ On ∧ 𝐵 ∈ On)) → ((𝐶 ·𝑜 𝐴) ∈ On ∧ (𝐶 ·𝑜 𝐵) ∈ On))
1312ancoms 468 . . . . 5 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐶 ∈ On) → ((𝐶 ·𝑜 𝐴) ∈ On ∧ (𝐶 ·𝑜 𝐵) ∈ On))
14133impa 1278 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐶 ·𝑜 𝐴) ∈ On ∧ (𝐶 ·𝑜 𝐵) ∈ On))
15 onsseleq 5803 . . . 4 (((𝐶 ·𝑜 𝐴) ∈ On ∧ (𝐶 ·𝑜 𝐵) ∈ On) → ((𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵) ↔ ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) ∨ (𝐶 ·𝑜 𝐴) = (𝐶 ·𝑜 𝐵))))
1614, 15syl 17 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵) ↔ ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) ∨ (𝐶 ·𝑜 𝐴) = (𝐶 ·𝑜 𝐵))))
1716adantr 480 . 2 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → ((𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵) ↔ ((𝐶 ·𝑜 𝐴) ∈ (𝐶 ·𝑜 𝐵) ∨ (𝐶 ·𝑜 𝐴) = (𝐶 ·𝑜 𝐵))))
186, 9, 173bitr4d 300 1 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 ↔ (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383  w3a 1054   = wceq 1523  wcel 2030  wss 3607  c0 3948  Oncon0 5761  (class class class)co 6690   ·𝑜 comu 7603
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-oadd 7609  df-omul 7610
This theorem is referenced by:  omwordi  7696  omeulem2  7708  oeeui  7727
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