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Theorem omword1 7540
 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 5650 . . . . 5 (𝐵 ∈ On → Ord 𝐵)
2 ordgt0ge1 7464 . . . . 5 (Ord 𝐵 → (∅ ∈ 𝐵 ↔ 1𝑜𝐵))
31, 2syl 17 . . . 4 (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ 1𝑜𝐵))
43adantl 481 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ 𝐵 ↔ 1𝑜𝐵))
5 1on 7454 . . . . . 6 1𝑜 ∈ On
6 omwordi 7538 . . . . . 6 ((1𝑜 ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On) → (1𝑜𝐵 → (𝐴 ·𝑜 1𝑜) ⊆ (𝐴 ·𝑜 𝐵)))
75, 6mp3an1 1403 . . . . 5 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (1𝑜𝐵 → (𝐴 ·𝑜 1𝑜) ⊆ (𝐴 ·𝑜 𝐵)))
87ancoms 468 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1𝑜𝐵 → (𝐴 ·𝑜 1𝑜) ⊆ (𝐴 ·𝑜 𝐵)))
9 om1 7509 . . . . . 6 (𝐴 ∈ On → (𝐴 ·𝑜 1𝑜) = 𝐴)
109adantr 480 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·𝑜 1𝑜) = 𝐴)
1110sseq1d 3595 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·𝑜 1𝑜) ⊆ (𝐴 ·𝑜 𝐵) ↔ 𝐴 ⊆ (𝐴 ·𝑜 𝐵)))
128, 11sylibd 228 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1𝑜𝐵𝐴 ⊆ (𝐴 ·𝑜 𝐵)))
134, 12sylbid 229 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ 𝐵𝐴 ⊆ (𝐴 ·𝑜 𝐵)))
1413imp 444 1 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → 𝐴 ⊆ (𝐴 ·𝑜 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540  ∅c0 3874  Ord word 5639  Oncon0 5640  (class class class)co 6549  1𝑜c1o 7440   ·𝑜 comu 7445 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-omul 7452 This theorem is referenced by:  om00  7542  cantnflem3  8471  cantnflem4  8472  cnfcomlem  8479
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