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Theorem omwordi 8608
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 8607 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 ↔ (𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵)))
21biimpd 229 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 → (𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵)))
32ex 412 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 → (𝐴𝐵 → (𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵))))
4 eloni 6396 . . . . . 6 (𝐶 ∈ On → Ord 𝐶)
5 ord0eln0 6441 . . . . . . 7 (Ord 𝐶 → (∅ ∈ 𝐶𝐶 ≠ ∅))
65necon2bbid 2982 . . . . . 6 (Ord 𝐶 → (𝐶 = ∅ ↔ ¬ ∅ ∈ 𝐶))
74, 6syl 17 . . . . 5 (𝐶 ∈ On → (𝐶 = ∅ ↔ ¬ ∅ ∈ 𝐶))
873ad2ant3 1134 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐶 = ∅ ↔ ¬ ∅ ∈ 𝐶))
9 ssid 4018 . . . . . . 7 ∅ ⊆ ∅
10 om0r 8576 . . . . . . . . 9 (𝐴 ∈ On → (∅ ·o 𝐴) = ∅)
1110adantr 480 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ·o 𝐴) = ∅)
12 om0r 8576 . . . . . . . . 9 (𝐵 ∈ On → (∅ ·o 𝐵) = ∅)
1312adantl 481 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ·o 𝐵) = ∅)
1411, 13sseq12d 4029 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((∅ ·o 𝐴) ⊆ (∅ ·o 𝐵) ↔ ∅ ⊆ ∅))
159, 14mpbiri 258 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ·o 𝐴) ⊆ (∅ ·o 𝐵))
16 oveq1 7438 . . . . . . 7 (𝐶 = ∅ → (𝐶 ·o 𝐴) = (∅ ·o 𝐴))
17 oveq1 7438 . . . . . . 7 (𝐶 = ∅ → (𝐶 ·o 𝐵) = (∅ ·o 𝐵))
1816, 17sseq12d 4029 . . . . . 6 (𝐶 = ∅ → ((𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵) ↔ (∅ ·o 𝐴) ⊆ (∅ ·o 𝐵)))
1915, 18syl5ibrcom 247 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐶 = ∅ → (𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵)))
20193adant3 1131 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐶 = ∅ → (𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵)))
218, 20sylbird 260 . . 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 179 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 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wss 3963  c0 4339  Ord word 6385  Oncon0 6386  (class class class)co 7431   ·o comu 8503
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-oadd 8509  df-omul 8510
This theorem is referenced by:  omword1  8610  omass  8617  omeulem1  8619  oewordri  8629  oeoalem  8633  oeeui  8639  oaabs2  8686  omxpenlem  9112  cantnflt  9710  cantnflem1d  9726  omabs2  43322  naddwordnexlem0  43386  oaltom  43395
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