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Theorem omword1 7811
Description: An ordinal is less than or equal to its product with another. (Contributed by NM, 21-Dec-2004.)
Assertion
Ref Expression
omword1 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → 𝐴 ⊆ (𝐴 ·𝑜 𝐵))

Proof of Theorem omword1
StepHypRef Expression
1 eloni 5875 . . . . 5 (𝐵 ∈ On → Ord 𝐵)
2 ordgt0ge1 7735 . . . . 5 (Ord 𝐵 → (∅ ∈ 𝐵 ↔ 1𝑜𝐵))
31, 2syl 17 . . . 4 (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ 1𝑜𝐵))
43adantl 467 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ 𝐵 ↔ 1𝑜𝐵))
5 1on 7724 . . . . . 6 1𝑜 ∈ On
6 omwordi 7809 . . . . . 6 ((1𝑜 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On) → (1𝑜𝐵 → (𝐴 ·𝑜 1𝑜) ⊆ (𝐴 ·𝑜 𝐵)))
75, 6mp3an1 1559 . . . . 5 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (1𝑜𝐵 → (𝐴 ·𝑜 1𝑜) ⊆ (𝐴 ·𝑜 𝐵)))
87ancoms 446 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1𝑜𝐵 → (𝐴 ·𝑜 1𝑜) ⊆ (𝐴 ·𝑜 𝐵)))
9 om1 7780 . . . . . 6 (𝐴 ∈ On → (𝐴 ·𝑜 1𝑜) = 𝐴)
109adantr 466 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 1𝑜) = 𝐴)
1110sseq1d 3781 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·𝑜 1𝑜) ⊆ (𝐴 ·𝑜 𝐵) ↔ 𝐴 ⊆ (𝐴 ·𝑜 𝐵)))
128, 11sylibd 229 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1𝑜𝐵𝐴 ⊆ (𝐴 ·𝑜 𝐵)))
134, 12sylbid 230 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ 𝐵𝐴 ⊆ (𝐴 ·𝑜 𝐵)))
1413imp 393 1 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → 𝐴 ⊆ (𝐴 ·𝑜 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  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:  om00  7813  cantnflem3  8756  cantnflem4  8757  cnfcomlem  8764
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