Theorem omword2 8183

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 8152	. . 3 ⊢ (𝐴 ∈ On → (1o ·o 𝐴) = 𝐴)
2	1	ad2antrr 725	. 2 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → (1o ·o 𝐴) = 𝐴)
3		eloni 6169	. . . . 5 ⊢ (𝐵 ∈ On → Ord 𝐵)
4		ordgt0ge1 8105	. . . . . 6 ⊢ (Ord 𝐵 → (∅ ∈ 𝐵 ↔ 1o ⊆ 𝐵))
5	4	biimpa 480	. . . . 5 ⊢ ((Ord 𝐵 ∧ ∅ ∈ 𝐵) → 1o ⊆ 𝐵)
6	3, 5	sylan 583	. . . 4 ⊢ ((𝐵 ∈ On ∧ ∅ ∈ 𝐵) → 1o ⊆ 𝐵)
7	6	adantll 713	. . 3 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → 1o ⊆ 𝐵)
8		1on 8092	. . . . . 6 ⊢ 1o ∈ On
9		omwordri 8181	. . . . . 6 ⊢ ((1o ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On) → (1o ⊆ 𝐵 → (1o ·o 𝐴) ⊆ (𝐵 ·o 𝐴)))
10	8, 9	mp3an1 1445	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (1o ⊆ 𝐵 → (1o ·o 𝐴) ⊆ (𝐵 ·o 𝐴)))
11	10	ancoms 462	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1o ⊆ 𝐵 → (1o ·o 𝐴) ⊆ (𝐵 ·o 𝐴)))
12	11	adantr 484	. . 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 3954	1 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → 𝐴 ⊆ (𝐵 ·o 𝐴))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ⊆ wss 3881  ∅c0 4243  Ord word 6158  Oncon0 6159  (class class class)co 7135  1_oc1o 8078   ·_o comu 8083

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

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 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-omul 8090

This theorem is referenced by:  omeulem1  8191  omabslem  8256  omabs  8257