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Theorem omword2 8185
Description: An ordinal is less than or equal to its product with another. (Contributed by NM, 21-Dec-2004.)
Assertion
Ref Expression
omword2 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → 𝐴 ⊆ (𝐵 ·o 𝐴))

Proof of Theorem omword2
StepHypRef Expression
1 om1r 8154 . . 3 (𝐴 ∈ On → (1o ·o 𝐴) = 𝐴)
21ad2antrr 725 . 2 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → (1o ·o 𝐴) = 𝐴)
3 eloni 6184 . . . . 5 (𝐵 ∈ On → Ord 𝐵)
4 ordgt0ge1 8107 . . . . . 6 (Ord 𝐵 → (∅ ∈ 𝐵 ↔ 1o𝐵))
54biimpa 480 . . . . 5 ((Ord 𝐵 ∧ ∅ ∈ 𝐵) → 1o𝐵)
63, 5sylan 583 . . . 4 ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → 1o𝐵)
76adantll 713 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → 1o𝐵)
8 1on 8094 . . . . . 6 1o ∈ On
9 omwordri 8183 . . . . . 6 ((1o ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On) → (1o𝐵 → (1o ·o 𝐴) ⊆ (𝐵 ·o 𝐴)))
108, 9mp3an1 1445 . . . . 5 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (1o𝐵 → (1o ·o 𝐴) ⊆ (𝐵 ·o 𝐴)))
1110ancoms 462 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1o𝐵 → (1o ·o 𝐴) ⊆ (𝐵 ·o 𝐴)))
1211adantr 484 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → (1o𝐵 → (1o ·o 𝐴) ⊆ (𝐵 ·o 𝐴)))
137, 12mpd 15 . 2 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → (1o ·o 𝐴) ⊆ (𝐵 ·o 𝐴))
142, 13eqsstrrd 3990 1 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → 𝐴 ⊆ (𝐵 ·o 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  wss 3918  c0 4274  Ord word 6173  Oncon0 6174  (class class class)co 7140  1oc1o 8080   ·o comu 8085
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5173  ax-sep 5186  ax-nul 5193  ax-pow 5249  ax-pr 5313  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-ral 3137  df-rex 3138  df-reu 3139  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-pss 3937  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-tp 4553  df-op 4555  df-uni 4822  df-iun 4904  df-br 5050  df-opab 5112  df-mpt 5130  df-tr 5156  df-id 5443  df-eprel 5448  df-po 5457  df-so 5458  df-fr 5497  df-we 5499  df-xp 5544  df-rel 5545  df-cnv 5546  df-co 5547  df-dm 5548  df-rn 5549  df-res 5550  df-ima 5551  df-pred 6131  df-ord 6177  df-on 6178  df-lim 6179  df-suc 6180  df-iota 6297  df-fun 6340  df-fn 6341  df-f 6342  df-f1 6343  df-fo 6344  df-f1o 6345  df-fv 6346  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-wrecs 7932  df-recs 7993  df-rdg 8031  df-1o 8087  df-oadd 8091  df-omul 8092
This theorem is referenced by:  omeulem1  8193  omabslem  8258  omabs  8259
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