Theorem omword1 8499

Description: An ordinal is less than or equal to its product with another. Lemma 3.11 of [Schloeder] p. 8. (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 6321	. . . . 5 ⊢ (𝐵 ∈ On → Ord 𝐵)
2		ordgt0ge1 8419	. . . . 5 ⊢ (Ord 𝐵 → (∅ ∈ 𝐵 ↔ 1o ⊆ 𝐵))
3	1, 2	syl 17	. . . 4 ⊢ (𝐵 ∈ On → (∅ ∈ 𝐵 ↔ 1o ⊆ 𝐵))
4	3	adantl 482	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ 𝐵 ↔ 1o ⊆ 𝐵))
5		1on 8408	. . . . . 6 ⊢ 1o ∈ On
6		omwordi 8497	. . . . . 6 ⊢ ((1o ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On) → (1o ⊆ 𝐵 → (𝐴 ·o 1o) ⊆ (𝐴 ·o 𝐵)))
7	5, 6	mp3an1 1456	. . . . 5 ⊢ ((𝐵 ∈ On ∧ 𝐴 ∈ On) → (1o ⊆ 𝐵 → (𝐴 ·o 1o) ⊆ (𝐴 ·o 𝐵)))
8	7	ancoms 459	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1o ⊆ 𝐵 → (𝐴 ·o 1o) ⊆ (𝐴 ·o 𝐵)))
9		om1 8468	. . . . . 6 ⊢ (𝐴 ∈ On → (𝐴 ·o 1o) = 𝐴)
10	9	adantr 481	. . . . 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 240	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1o ⊆ 𝐵 → 𝐴 ⊆ (𝐴 ·o 𝐵)))
13	4, 12	sylbid 241	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∅ ∈ 𝐵 → 𝐴 ⊆ (𝐴 ·o 𝐵)))
14	13	imp 407	1 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ ∅ ∈ 𝐵) → 𝐴 ⊆ (𝐴 ·o 𝐵))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119   ⊆ wss 3883  ∅c0 4262  Ord word 6310  Oncon0 6311  (class class class)co 7357  1_oc1o 8389   ·_o comu 8394

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5200  ax-sep 5219  ax-nul 5229  ax-pr 5363  ax-un 7679

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4263  df-if 4456  df-pw 4532  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4840  df-iun 4924  df-br 5074  df-opab 5136  df-mpt 5155  df-tr 5181  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7808  df-2nd 7933  df-frecs 8222  df-wrecs 8253  df-recs 8302  df-rdg 8340  df-1o 8396  df-oadd 8400  df-omul 8401

This theorem is referenced by:  om00  8501  cantnflem3  9604  cantnflem4  9605  cnfcomlem  9612  omge1  43751  cantnftermord  43774  naddwordnexlem4  43855