Theorem omword1 8182

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

Step	Hyp	Ref	Expression
1		eloni 6169	. . . . 5 ⊢ (𝐵 ∈ On → Ord 𝐵)
2		ordgt0ge1 8105	. . . . 5 ⊢ (Ord 𝐵 → (∅ ∈ 𝐵 ↔ 1o ⊆ 𝐵))
3	1, 2	syl 17	. . . 4 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ 1o ⊆ 𝐵))
4	3	adantl 485	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ 𝐵 ↔ 1o ⊆ 𝐵))
5		1on 8092	. . . . . 6 ⊢ 1o ∈ On
6		omwordi 8180	. . . . . 6 ⊢ ((1o ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On) → (1o ⊆ 𝐵 → (𝐴 ·o 1o) ⊆ (𝐴 ·o 𝐵)))
7	5, 6	mp3an1 1445	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (1o ⊆ 𝐵 → (𝐴 ·o 1o) ⊆ (𝐴 ·o 𝐵)))
8	7	ancoms 462	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1o ⊆ 𝐵 → (𝐴 ·o 1o) ⊆ (𝐴 ·o 𝐵)))
9		om1 8151	. . . . . 6 ⊢ (𝐴 ∈ On → (𝐴 ·o 1o) = 𝐴)
10	9	adantr 484	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 1o) = 𝐴)
11	10	sseq1d 3946	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o 1o) ⊆ (𝐴 ·o 𝐵) ↔ 𝐴 ⊆ (𝐴 ·o 𝐵)))
12	8, 11	sylibd 242	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1o ⊆ 𝐵 → 𝐴 ⊆ (𝐴 ·o 𝐵)))
13	4, 12	sylbid 243	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ 𝐵 → 𝐴 ⊆ (𝐴 ·o 𝐵)))
14	13	imp 410	1 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → 𝐴 ⊆ (𝐴 ·o 𝐵))

Syntax hints:   → wi 4   ↔ wb 209   ∧ 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:  om00  8184  cantnflem3  9138  cantnflem4  9139  cnfcomlem  9146