Theorem onexomgt 43258

Description: For any ordinal, there is always a larger product of omega. (Contributed by RP, 1-Feb-2025.)

Assertion

Ref	Expression
onexomgt	⊢ (𝐴 ∈ On → ∃𝑥 ∈ On 𝐴 ∈ (ω ·o 𝑥))

Distinct variable group:   𝑥,𝐴

Proof of Theorem onexomgt

Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		omelon 9687	. . 3 ⊢ ω ∈ On
2		peano1 7911	. . . 4 ⊢ ∅ ∈ ω
3	2	ne0ii 4343	. . 3 ⊢ ω ≠ ∅
4		omeu 8624	. . 3 ⊢ ((ω ∈ On ∧ 𝐴 ∈ On ∧ ω ≠ ∅) → ∃!𝑐∃𝑎 ∈ On ∃𝑏 ∈ ω (𝑐 = ⟨𝑎, 𝑏⟩ ∧ ((ω ·o 𝑎) +o 𝑏) = 𝐴))
5	1, 3, 4	mp3an13 1453	. 2 ⊢ (𝐴 ∈ On → ∃!𝑐∃𝑎 ∈ On ∃𝑏 ∈ ω (𝑐 = ⟨𝑎, 𝑏⟩ ∧ ((ω ·o 𝑎) +o 𝑏) = 𝐴))
6		euex 2576	. . 3 ⊢ (∃!𝑐∃𝑎 ∈ On ∃𝑏 ∈ ω (𝑐 = ⟨𝑎, 𝑏⟩ ∧ ((ω ·o 𝑎) +o 𝑏) = 𝐴) → ∃𝑐∃𝑎 ∈ On ∃𝑏 ∈ ω (𝑐 = ⟨𝑎, 𝑏⟩ ∧ ((ω ·o 𝑎) +o 𝑏) = 𝐴))
7		onsuc 7832	. . . . . . . . . 10 ⊢ (𝑎 ∈ On → suc 𝑎 ∈ On)
8	7	adantr 480	. . . . . . . . 9 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ ω) → suc 𝑎 ∈ On)
9		simpr 484	. . . . . . . . . 10 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ ω) ∧ ((ω ·o 𝑎) +o 𝑏) = 𝐴) → ((ω ·o 𝑎) +o 𝑏) = 𝐴)
10		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ ω) → 𝑏 ∈ ω)
11		simpl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ ω) → 𝑎 ∈ On)
12		omcl 8575	. . . . . . . . . . . . . . 15 ⊢ ((ω ∈ On ∧ 𝑎 ∈ On) → (ω ·o 𝑎) ∈ On)
13	1, 11, 12	sylancr 587	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ ω) → (ω ·o 𝑎) ∈ On)
14		oaordi 8585	. . . . . . . . . . . . . 14 ⊢ ((ω ∈ On ∧ (ω ·o 𝑎) ∈ On) → (𝑏 ∈ ω → ((ω ·o 𝑎) +o 𝑏) ∈ ((ω ·o 𝑎) +o ω)))
15	1, 13, 14	sylancr 587	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ ω) → (𝑏 ∈ ω → ((ω ·o 𝑎) +o 𝑏) ∈ ((ω ·o 𝑎) +o ω)))
16	10, 15	mpd 15	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ ω) → ((ω ·o 𝑎) +o 𝑏) ∈ ((ω ·o 𝑎) +o ω))
17		omsuc 8565	. . . . . . . . . . . . 13 ⊢ ((ω ∈ On ∧ 𝑎 ∈ On) → (ω ·o suc 𝑎) = ((ω ·o 𝑎) +o ω))
18	1, 11, 17	sylancr 587	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ ω) → (ω ·o suc 𝑎) = ((ω ·o 𝑎) +o ω))
19	16, 18	eleqtrrd 2843	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ ω) → ((ω ·o 𝑎) +o 𝑏) ∈ (ω ·o suc 𝑎))
20	19	adantr 480	. . . . . . . . . 10 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ ω) ∧ ((ω ·o 𝑎) +o 𝑏) = 𝐴) → ((ω ·o 𝑎) +o 𝑏) ∈ (ω ·o suc 𝑎))
21	9, 20	eqeltrrd 2841	. . . . . . . . 9 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ ω) ∧ ((ω ·o 𝑎) +o 𝑏) = 𝐴) → 𝐴 ∈ (ω ·o suc 𝑎))
22		oveq2 7440	. . . . . . . . . . 11 ⊢ (𝑥 = suc 𝑎 → (ω ·o 𝑥) = (ω ·o suc 𝑎))
23	22	eleq2d 2826	. . . . . . . . . 10 ⊢ (𝑥 = suc 𝑎 → (𝐴 ∈ (ω ·o 𝑥) ↔ 𝐴 ∈ (ω ·o suc 𝑎)))
24	23	rspcev 3621	. . . . . . . . 9 ⊢ ((suc 𝑎 ∈ On ∧ 𝐴 ∈ (ω ·o suc 𝑎)) → ∃𝑥 ∈ On 𝐴 ∈ (ω ·o 𝑥))
25	8, 21, 24	syl2an2r 685	. . . . . . . 8 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ ω) ∧ ((ω ·o 𝑎) +o 𝑏) = 𝐴) → ∃𝑥 ∈ On 𝐴 ∈ (ω ·o 𝑥))
26	25	ex 412	. . . . . . 7 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ ω) → (((ω ·o 𝑎) +o 𝑏) = 𝐴 → ∃𝑥 ∈ On 𝐴 ∈ (ω ·o 𝑥)))
27	26	adantld 490	. . . . . 6 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ ω) → ((𝑐 = ⟨𝑎, 𝑏⟩ ∧ ((ω ·o 𝑎) +o 𝑏) = 𝐴) → ∃𝑥 ∈ On 𝐴 ∈ (ω ·o 𝑥)))
28	27	a1i 11	. . . . 5 ⊢ (𝐴 ∈ On → ((𝑎 ∈ On ∧ 𝑏 ∈ ω) → ((𝑐 = ⟨𝑎, 𝑏⟩ ∧ ((ω ·o 𝑎) +o 𝑏) = 𝐴) → ∃𝑥 ∈ On 𝐴 ∈ (ω ·o 𝑥))))
29	28	rexlimdvv 3211	. . . 4 ⊢ (𝐴 ∈ On → (∃𝑎 ∈ On ∃𝑏 ∈ ω (𝑐 = ⟨𝑎, 𝑏⟩ ∧ ((ω ·o 𝑎) +o 𝑏) = 𝐴) → ∃𝑥 ∈ On 𝐴 ∈ (ω ·o 𝑥)))
30	29	exlimdv 1932	. . 3 ⊢ (𝐴 ∈ On → (∃𝑐∃𝑎 ∈ On ∃𝑏 ∈ ω (𝑐 = ⟨𝑎, 𝑏⟩ ∧ ((ω ·o 𝑎) +o 𝑏) = 𝐴) → ∃𝑥 ∈ On 𝐴 ∈ (ω ·o 𝑥)))
31	6, 30	syl5 34	. 2 ⊢ (𝐴 ∈ On → (∃!𝑐∃𝑎 ∈ On ∃𝑏 ∈ ω (𝑐 = ⟨𝑎, 𝑏⟩ ∧ ((ω ·o 𝑎) +o 𝑏) = 𝐴) → ∃𝑥 ∈ On 𝐴 ∈ (ω ·o 𝑥)))
32	5, 31	mpd 15	1 ⊢ (𝐴 ∈ On → ∃𝑥 ∈ On 𝐴 ∈ (ω ·o 𝑥))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539  ∃wex 1778   ∈ wcel 2107  ∃!weu 2567   ≠ wne 2939  ∃wrex 3069  ∅c0 4332  ⟨cop 4631  Oncon0 6383  suc csuc 6385  (class class class)co 7432  ωcom 7888   +_o coa 8504   ·_o comu 8505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756  ax-inf2 9682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-1o 8507  df-oadd 8511  df-omul 8512

This theorem is referenced by:  onexlimgt  43260