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Theorem omword1 8197
 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 6190 . . . . 5 (𝐵 ∈ On → Ord 𝐵)
2 ordgt0ge1 8120 . . . . 5 (Ord 𝐵 → (∅ ∈ 𝐵 ↔ 1o𝐵))
31, 2syl 17 . . . 4 (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ 1o𝐵))
43adantl 485 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ 𝐵 ↔ 1o𝐵))
5 1on 8107 . . . . . 6 1o ∈ On
6 omwordi 8195 . . . . . 6 ((1o ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On) → (1o𝐵 → (𝐴 ·o 1o) ⊆ (𝐴 ·o 𝐵)))
75, 6mp3an1 1445 . . . . 5 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (1o𝐵 → (𝐴 ·o 1o) ⊆ (𝐴 ·o 𝐵)))
87ancoms 462 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1o𝐵 → (𝐴 ·o 1o) ⊆ (𝐴 ·o 𝐵)))
9 om1 8166 . . . . . 6 (𝐴 ∈ On → (𝐴 ·o 1o) = 𝐴)
109adantr 484 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 1o) = 𝐴)
1110sseq1d 3984 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o 1o) ⊆ (𝐴 ·o 𝐵) ↔ 𝐴 ⊆ (𝐴 ·o 𝐵)))
128, 11sylibd 242 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1o𝐵𝐴 ⊆ (𝐴 ·o 𝐵)))
134, 12sylbid 243 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ 𝐵𝐴 ⊆ (𝐴 ·o 𝐵)))
1413imp 410 1 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → 𝐴 ⊆ (𝐴 ·o 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2115   ⊆ wss 3919  ∅c0 4276  Ord word 6179  Oncon0 6180  (class class class)co 7151  1oc1o 8093   ·o comu 8098 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 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7457 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 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-iun 4907  df-br 5054  df-opab 5116  df-mpt 5134  df-tr 5160  df-id 5448  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-we 5504  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-pred 6137  df-ord 6183  df-on 6184  df-lim 6185  df-suc 6186  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-f1 6350  df-fo 6351  df-f1o 6352  df-fv 6353  df-ov 7154  df-oprab 7155  df-mpo 7156  df-om 7577  df-wrecs 7945  df-recs 8006  df-rdg 8044  df-1o 8100  df-oadd 8104  df-omul 8105 This theorem is referenced by:  om00  8199  cantnflem3  9153  cantnflem4  9154  cnfcomlem  9161
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