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Theorem omwordi 8577
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 8576 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 ↔ (𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵)))
21biimpd 228 . . 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 6374 . . . . . 6 (𝐶 ∈ On → Ord 𝐶)
5 ord0eln0 6419 . . . . . . 7 (Ord 𝐶 → (∅ ∈ 𝐶𝐶 ≠ ∅))
65necon2bbid 2983 . . . . . 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 4004 . . . . . . 7 ∅ ⊆ ∅
10 om0r 8545 . . . . . . . . 9 (𝐴 ∈ On → (∅ ·o 𝐴) = ∅)
1110adantr 480 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ·o 𝐴) = ∅)
12 om0r 8545 . . . . . . . . 9 (𝐵 ∈ On → (∅ ·o 𝐵) = ∅)
1312adantl 481 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ·o 𝐵) = ∅)
1411, 13sseq12d 4015 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((∅ ·o 𝐴) ⊆ (∅ ·o 𝐵) ↔ ∅ ⊆ ∅))
159, 14mpbiri 258 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ·o 𝐴) ⊆ (∅ ·o 𝐵))
16 oveq1 7419 . . . . . . 7 (𝐶 = ∅ → (𝐶 ·o 𝐴) = (∅ ·o 𝐴))
17 oveq1 7419 . . . . . . 7 (𝐶 = ∅ → (𝐶 ·o 𝐵) = (∅ ·o 𝐵))
1816, 17sseq12d 4015 . . . . . 6 (𝐶 = ∅ → ((𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵) ↔ (∅ ·o 𝐴) ⊆ (∅ ·o 𝐵)))
1915, 18syl5ibrcom 246 . . . . 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 205  wa 395  w3a 1086   = wceq 1540  wcel 2105  wss 3948  c0 4322  Ord word 6363  Oncon0 6364  (class class class)co 7412   ·o comu 8470
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7860  df-2nd 7980  df-frecs 8272  df-wrecs 8303  df-recs 8377  df-rdg 8416  df-oadd 8476  df-omul 8477
This theorem is referenced by:  omword1  8579  omass  8586  omeulem1  8588  oewordri  8598  oeoalem  8602  oeeui  8608  oaabs2  8654  omxpenlem  9079  cantnflt  9673  cantnflem1d  9689  omabs2  42545  naddwordnexlem0  42610  oaltom  42619
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