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Theorem omword2 7812
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 7781 . . 3 (𝐴 ∈ On → (1𝑜 ·𝑜 𝐴) = 𝐴)
21ad2antrr 705 . 2 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → (1𝑜 ·𝑜 𝐴) = 𝐴)
3 eloni 5875 . . . . 5 (𝐵 ∈ On → Ord 𝐵)
4 ordgt0ge1 7735 . . . . . 6 (Ord 𝐵 → (∅ ∈ 𝐵 ↔ 1𝑜𝐵))
54biimpa 462 . . . . 5 ((Ord 𝐵 ∧ ∅ ∈ 𝐵) → 1𝑜𝐵)
63, 5sylan 569 . . . 4 ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → 1𝑜𝐵)
76adantll 693 . . 3 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → 1𝑜𝐵)
8 1on 7724 . . . . . 6 1𝑜 ∈ On
9 omwordri 7810 . . . . . 6 ((1𝑜 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On) → (1𝑜𝐵 → (1𝑜 ·𝑜 𝐴) ⊆ (𝐵 ·𝑜 𝐴)))
108, 9mp3an1 1559 . . . . 5 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (1𝑜𝐵 → (1𝑜 ·𝑜 𝐴) ⊆ (𝐵 ·𝑜 𝐴)))
1110ancoms 446 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1𝑜𝐵 → (1𝑜 ·𝑜 𝐴) ⊆ (𝐵 ·𝑜 𝐴)))
1211adantr 466 . . 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, 13eqsstr3d 3789 1 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → 𝐴 ⊆ (𝐵 ·𝑜 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wss 3723  c0 4063  Ord word 5864  Oncon0 5865  (class class class)co 6796  1𝑜c1o 7710   ·𝑜 comu 7715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-om 7217  df-1st 7319  df-2nd 7320  df-wrecs 7563  df-recs 7625  df-rdg 7663  df-1o 7717  df-oadd 7721  df-omul 7722
This theorem is referenced by:  omeulem1  7820  omabslem  7884  omabs  7885
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