Theorem omword2 8405

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

Step	Hyp	Ref	Expression
1		om1r 8374	. . 3 ⊢ (𝐴 ∈ On → (1o ·o 𝐴) = 𝐴)
2	1	ad2antrr 723	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → (1o ·o 𝐴) = 𝐴)
3		eloni 6276	. . . . 5 ⊢ (𝐵 ∈ On → Ord 𝐵)
4		ordgt0ge1 8323	. . . . . 6 ⊢ (Ord 𝐵 → (∅ ∈ 𝐵 ↔ 1o ⊆ 𝐵))
5	4	biimpa 477	. . . . 5 ⊢ ((Ord 𝐵 ∧ ∅ ∈ 𝐵) → 1o ⊆ 𝐵)
6	3, 5	sylan 580	. . . 4 ⊢ ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → 1o ⊆ 𝐵)
7	6	adantll 711	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → 1o ⊆ 𝐵)
8		1on 8309	. . . . . 6 ⊢ 1o ∈ On
9		omwordri 8403	. . . . . 6 ⊢ ((1o ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On) → (1o ⊆ 𝐵 → (1o ·o 𝐴) ⊆ (𝐵 ·o 𝐴)))
10	8, 9	mp3an1 1447	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (1o ⊆ 𝐵 → (1o ·o 𝐴) ⊆ (𝐵 ·o 𝐴)))
11	10	ancoms 459	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1o ⊆ 𝐵 → (1o ·o 𝐴) ⊆ (𝐵 ·o 𝐴)))
12	11	adantr 481	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → (1o ⊆ 𝐵 → (1o ·o 𝐴) ⊆ (𝐵 ·o 𝐴)))
13	7, 12	mpd 15	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → (1o ·o 𝐴) ⊆ (𝐵 ·o 𝐴))
14	2, 13	eqsstrrd 3960	1 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → 𝐴 ⊆ (𝐵 ·o 𝐴))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ⊆ wss 3887  ∅c0 4256  Ord word 6265  Oncon0 6266  (class class class)co 7275  1_oc1o 8290   ·_o comu 8295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-oadd 8301  df-omul 8302

This theorem is referenced by:  omeulem1  8413  omabslem  8480  omabs  8481