Theorem omword1 8543

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 6357	. . . . 5 ⊢ (𝐵 ∈ On → Ord 𝐵)
2		ordgt0ge1 8463	. . . . 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 8451	. . . . . 6 ⊢ 1o ∈ On
6		omwordi 8541	. . . . . 6 ⊢ ((1o ∈ On ∧ 𝐵 ∈ On ∧ 𝐴 ∈ On) → (1o ⊆ 𝐵 → (𝐴 ·o 1o) ⊆ (𝐴 ·o 𝐵)))
7	5, 6	mp3an1 1470	. . . . 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 8512	. . . . . 6 ⊢ (𝐴 ∈ On → (𝐴 ·o 1o) = 𝐴)
10	9	adantr 484	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ·o 1o) = 𝐴)
11	10	sseq1d 3968	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ·o 1o) ⊆ (𝐴 ·o 𝐵) ↔ 𝐴 ⊆ (𝐴 ·o 𝐵)))
12	8, 11	sylibd 241	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (1o ⊆ 𝐵 → 𝐴 ⊆ (𝐴 ·o 𝐵)))
13	4, 12	sylbid 242	. 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 208   ∧ wa 399   = wceq 1561   ∈ wcel 2143   ⊆ wss 3905  ∅c0 4286  Ord word 6346  Oncon0 6347  (class class class)co 7397  1_oc1o 8431   ·_o comu 8436

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5228  ax-sep 5247  ax-nul 5257  ax-pr 5391  ax-un 7719

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3078  df-rex 3088  df-reu 3369  df-rab 3416  df-v 3457  df-sbc 3746  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5102  df-opab 5164  df-mpt 5183  df-tr 5209  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-res 5660  df-ima 5661  df-pred 6289  df-ord 6350  df-on 6351  df-lim 6352  df-suc 6353  df-iota 6478  df-fun 6524  df-fn 6525  df-f 6526  df-f1 6527  df-fo 6528  df-f1o 6529  df-fv 6530  df-ov 7400  df-oprab 7401  df-mpo 7402  df-om 7848  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8382  df-1o 8438  df-oadd 8442  df-omul 8443

This theorem is referenced by:  om00  8545  cantnflem3  9647  cantnflem4  9648  cnfcomlem  9655  omge1  43875  cantnftermord  43898  naddwordnexlem4  43979