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Theorem omwordi 8228
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 8227 . . . 4 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 ↔ (𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵)))
21biimpd 232 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) ∧ ∅ ∈ 𝐶) → (𝐴𝐵 → (𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵)))
32ex 416 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (∅ ∈ 𝐶 → (𝐴𝐵 → (𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵))))
4 eloni 6182 . . . . . 6 (𝐶 ∈ On → Ord 𝐶)
5 ord0eln0 6226 . . . . . . 7 (Ord 𝐶 → (∅ ∈ 𝐶𝐶 ≠ ∅))
65necon2bbid 2977 . . . . . 6 (Ord 𝐶 → (𝐶 = ∅ ↔ ¬ ∅ ∈ 𝐶))
74, 6syl 17 . . . . 5 (𝐶 ∈ On → (𝐶 = ∅ ↔ ¬ ∅ ∈ 𝐶))
873ad2ant3 1136 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐶 = ∅ ↔ ¬ ∅ ∈ 𝐶))
9 ssid 3899 . . . . . . 7 ∅ ⊆ ∅
10 om0r 8195 . . . . . . . . 9 (𝐴 ∈ On → (∅ ·o 𝐴) = ∅)
1110adantr 484 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ·o 𝐴) = ∅)
12 om0r 8195 . . . . . . . . 9 (𝐵 ∈ On → (∅ ·o 𝐵) = ∅)
1312adantl 485 . . . . . . . 8 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ·o 𝐵) = ∅)
1411, 13sseq12d 3910 . . . . . . 7 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((∅ ·o 𝐴) ⊆ (∅ ·o 𝐵) ↔ ∅ ⊆ ∅))
159, 14mpbiri 261 . . . . . 6 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ·o 𝐴) ⊆ (∅ ·o 𝐵))
16 oveq1 7177 . . . . . . 7 (𝐶 = ∅ → (𝐶 ·o 𝐴) = (∅ ·o 𝐴))
17 oveq1 7177 . . . . . . 7 (𝐶 = ∅ → (𝐶 ·o 𝐵) = (∅ ·o 𝐵))
1816, 17sseq12d 3910 . . . . . 6 (𝐶 = ∅ → ((𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵) ↔ (∅ ·o 𝐴) ⊆ (∅ ·o 𝐵)))
1915, 18syl5ibrcom 250 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐶 = ∅ → (𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵)))
20193adant3 1133 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → (𝐶 = ∅ → (𝐶 ·o 𝐴) ⊆ (𝐶 ·o 𝐵)))
218, 20sylbird 263 . . 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 182 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 209  wa 399  w3a 1088   = wceq 1542  wcel 2114  wss 3843  c0 4211  Ord word 6171  Oncon0 6172  (class class class)co 7170   ·o comu 8129
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pr 5296  ax-un 7479
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-ov 7173  df-oprab 7174  df-mpo 7175  df-om 7600  df-wrecs 7976  df-recs 8037  df-rdg 8075  df-oadd 8135  df-omul 8136
This theorem is referenced by:  omword1  8230  omass  8237  omeulem1  8239  oewordri  8249  oeoalem  8253  oeeui  8259  oaabs2  8303  omxpenlem  8667  cantnflt  9208  cantnflem1d  9224
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