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Theorem omword1 8002
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 6041 . . . . 5 (𝐵 ∈ On → Ord 𝐵)
2 ordgt0ge1 7926 . . . . 5 (Ord 𝐵 → (∅ ∈ 𝐵 ↔ 1o𝐵))
31, 2syl 17 . . . 4 (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ 1o𝐵))
43adantl 474 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ 𝐵 ↔ 1o𝐵))
5 1on 7914 . . . . . 6 1o ∈ On
6 omwordi 8000 . . . . . 6 ((1o ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On) → (1o𝐵 → (𝐴 ·o 1o) ⊆ (𝐴 ·o 𝐵)))
75, 6mp3an1 1427 . . . . 5 ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (1o𝐵 → (𝐴 ·o 1o) ⊆ (𝐴 ·o 𝐵)))
87ancoms 451 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1o𝐵 → (𝐴 ·o 1o) ⊆ (𝐴 ·o 𝐵)))
9 om1 7971 . . . . . 6 (𝐴 ∈ On → (𝐴 ·o 1o) = 𝐴)
109adantr 473 . . . . 5 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 1o) = 𝐴)
1110sseq1d 3890 . . . 4 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o 1o) ⊆ (𝐴 ·o 𝐵) ↔ 𝐴 ⊆ (𝐴 ·o 𝐵)))
128, 11sylibd 231 . . 3 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1o𝐵𝐴 ⊆ (𝐴 ·o 𝐵)))
134, 12sylbid 232 . 2 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ 𝐵𝐴 ⊆ (𝐴 ·o 𝐵)))
1413imp 398 1 (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → 𝐴 ⊆ (𝐴 ·o 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387   = wceq 1507  wcel 2050  wss 3831  c0 4180  Ord word 6030  Oncon0 6031  (class class class)co 6978  1oc1o 7900   ·o comu 7905
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5050  ax-sep 5061  ax-nul 5068  ax-pow 5120  ax-pr 5187  ax-un 7281
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rab 3097  df-v 3417  df-sbc 3684  df-csb 3789  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-pss 3847  df-nul 4181  df-if 4352  df-pw 4425  df-sn 4443  df-pr 4445  df-tp 4447  df-op 4449  df-uni 4714  df-iun 4795  df-br 4931  df-opab 4993  df-mpt 5010  df-tr 5032  df-id 5313  df-eprel 5318  df-po 5327  df-so 5328  df-fr 5367  df-we 5369  df-xp 5414  df-rel 5415  df-cnv 5416  df-co 5417  df-dm 5418  df-rn 5419  df-res 5420  df-ima 5421  df-pred 5988  df-ord 6034  df-on 6035  df-lim 6036  df-suc 6037  df-iota 6154  df-fun 6192  df-fn 6193  df-f 6194  df-f1 6195  df-fo 6196  df-f1o 6197  df-fv 6198  df-ov 6981  df-oprab 6982  df-mpo 6983  df-om 7399  df-wrecs 7752  df-recs 7814  df-rdg 7852  df-1o 7907  df-oadd 7911  df-omul 7912
This theorem is referenced by:  om00  8004  cantnflem3  8950  cantnflem4  8951  cnfcomlem  8958
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