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Theorem omwordi 7696
Description: Weak ordering property of ordinal multiplication. (Contributed by NM, 21-Dec-2004.)
Assertion
Ref Expression
omwordi ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵)))

Proof of Theorem omwordi
StepHypRef Expression
1 omword 7695 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 ↔ (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵)))
21biimpd 219 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 → (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵)))
32ex 449 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 → (𝐴𝐵 → (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵))))
4 eloni 5771 . . . . . 6 (𝐶 ∈ On → Ord 𝐶)
5 ord0eln0 5817 . . . . . . 7 (Ord 𝐶 → (∅ ∈ 𝐶𝐶 ≠ ∅))
65necon2bbid 2866 . . . . . 6 (Ord 𝐶 → (𝐶 = ∅ ↔ ¬ ∅ ∈ 𝐶))
74, 6syl 17 . . . . 5 (𝐶 ∈ On → (𝐶 = ∅ ↔ ¬ ∅ ∈ 𝐶))
873ad2ant3 1104 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐶 = ∅ ↔ ¬ ∅ ∈ 𝐶))
9 ssid 3657 . . . . . . 7 ∅ ⊆ ∅
10 om0r 7664 . . . . . . . . 9 (𝐴 ∈ On → (∅ ·𝑜 𝐴) = ∅)
1110adantr 480 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ·𝑜 𝐴) = ∅)
12 om0r 7664 . . . . . . . . 9 (𝐵 ∈ On → (∅ ·𝑜 𝐵) = ∅)
1312adantl 481 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ·𝑜 𝐵) = ∅)
1411, 13sseq12d 3667 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((∅ ·𝑜 𝐴) ⊆ (∅ ·𝑜 𝐵) ↔ ∅ ⊆ ∅))
159, 14mpbiri 248 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ·𝑜 𝐴) ⊆ (∅ ·𝑜 𝐵))
16 oveq1 6697 . . . . . . 7 (𝐶 = ∅ → (𝐶 ·𝑜 𝐴) = (∅ ·𝑜 𝐴))
17 oveq1 6697 . . . . . . 7 (𝐶 = ∅ → (𝐶 ·𝑜 𝐵) = (∅ ·𝑜 𝐵))
1816, 17sseq12d 3667 . . . . . 6 (𝐶 = ∅ → ((𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵) ↔ (∅ ·𝑜 𝐴) ⊆ (∅ ·𝑜 𝐵)))
1915, 18syl5ibrcom 237 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐶 = ∅ → (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵)))
20193adant3 1101 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐶 = ∅ → (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵)))
218, 20sylbird 250 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (¬ ∅ ∈ 𝐶 → (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵)))
2221a1dd 50 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (¬ ∅ ∈ 𝐶 → (𝐴𝐵 → (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵))))
233, 22pm2.61d 170 1 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐴𝐵 → (𝐶 ·𝑜 𝐴) ⊆ (𝐶 ·𝑜 𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wss 3607  c0 3948  Ord word 5760  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-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-oadd 7609  df-omul 7610
This theorem is referenced by:  omword1  7698  omass  7705  omeulem1  7707  oewordri  7717  oeoalem  7721  oeeui  7727  oaabs2  7770  omxpenlem  8102  cantnflt  8607  cantnflem1d  8623
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