Theorem onexomgt 44028

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 9622	. . 3 ⊢ ω ∈ On
2		peano1 7891	. . . 4 ⊢ ∅ ∈ ω
3	2	ne0ii 4297	. . 3 ⊢ ω ≠ ∅
4		omeu 8576	. . 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 1481	. 2 ⊢ (𝐴 ∈ On → ∃!𝑐∃𝑎 ∈ On ∃𝑏 ∈ ω (𝑐 = ⟨𝑎, 𝑏⟩ ∧ ((ω ·o 𝑎) +o 𝑏) = 𝐴))
6		euex 2607	. . 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 7815	. . . . . . . . . 10 ⊢ (𝑎 ∈ On → suc 𝑎 ∈ On)
8	7	adantr 486	. . . . . . . . 9 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ ω) → suc 𝑎 ∈ On)
9		simpr 490	. . . . . . . . . 10 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ ω) ∧ ((ω ·o 𝑎) +o 𝑏) = 𝐴) → ((ω ·o 𝑎) +o 𝑏) = 𝐴)
10		simpr 490	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ ω) → 𝑏 ∈ ω)
11		simpl 488	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ ω) → 𝑎 ∈ On)
12		omcl 8527	. . . . . . . . . . . . . . 15 ⊢ ((ω ∈ On ∧ 𝑎 ∈ On) → (ω ·o 𝑎) ∈ On)
13	1, 11, 12	sylancr 599	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ ω) → (ω ·o 𝑎) ∈ On)
14		oaordi 8537	. . . . . . . . . . . . . 14 ⊢ ((ω ∈ On ∧ (ω ·o 𝑎) ∈ On) → (𝑏 ∈ ω → ((ω ·o 𝑎) +o 𝑏) ∈ ((ω ·o 𝑎) +o ω)))
15	1, 13, 14	sylancr 599	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ ω) → (𝑏 ∈ ω → ((ω ·o 𝑎) +o 𝑏) ∈ ((ω ·o 𝑎) +o ω)))
16	10, 15	mpd 16	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ ω) → ((ω ·o 𝑎) +o 𝑏) ∈ ((ω ·o 𝑎) +o ω))
17		omsuc 8517	. . . . . . . . . . . . 13 ⊢ ((ω ∈ On ∧ 𝑎 ∈ On) → (ω ·o suc 𝑎) = ((ω ·o 𝑎) +o ω))
18	1, 11, 17	sylancr 599	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ ω) → (ω ·o suc 𝑎) = ((ω ·o 𝑎) +o ω))
19	16, 18	eleqtrrd 2868	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ ω) → ((ω ·o 𝑎) +o 𝑏) ∈ (ω ·o suc 𝑎))
20	19	adantr 486	. . . . . . . . . 10 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ ω) ∧ ((ω ·o 𝑎) +o 𝑏) = 𝐴) → ((ω ·o 𝑎) +o 𝑏) ∈ (ω ·o suc 𝑎))
21	9, 20	eqeltrrd 2866	. . . . . . . . 9 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ ω) ∧ ((ω ·o 𝑎) +o 𝑏) = 𝐴) → 𝐴 ∈ (ω ·o suc 𝑎))
22		oveq2 7427	. . . . . . . . . . 11 ⊢ (𝑥 = suc 𝑎 → (ω ·o 𝑥) = (ω ·o suc 𝑎))
23	22	eleq2d 2851	. . . . . . . . . 10 ⊢ (𝑥 = suc 𝑎 → (𝐴 ∈ (ω ·o 𝑥) ↔ 𝐴 ∈ (ω ·o suc 𝑎)))
24	23	rspcev 3583	. . . . . . . . 9 ⊢ ((suc 𝑎 ∈ On ∧ 𝐴 ∈ (ω ·o suc 𝑎)) → ∃𝑥 ∈ On 𝐴 ∈ (ω ·o 𝑥))
25	8, 21, 24	syl2an2r 698	. . . . . . . 8 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ ω) ∧ ((ω ·o 𝑎) +o 𝑏) = 𝐴) → ∃𝑥 ∈ On 𝐴 ∈ (ω ·o 𝑥))
26	25	ex 418	. . . . . . 7 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ ω) → (((ω ·o 𝑎) +o 𝑏) = 𝐴 → ∃𝑥 ∈ On 𝐴 ∈ (ω ·o 𝑥)))
27	26	adantld 496	. . . . . 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 3223	. . . 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 1966	. . 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 35	. 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 16	1 ⊢ (𝐴 ∈ On → ∃𝑥 ∈ On 𝐴 ∈ (ω ·o 𝑥))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2146  ∃!weu 2598   ≠ wne 2960  ∃wrex 3091  ∅c0 4286  ⟨cop 4597  Oncon0 6364  suc csuc 6366  (class class class)co 7419  ωcom 7868   +_o coa 8456   ·_o comu 8457

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7742  ax-inf2 9617

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-oadd 8463  df-omul 8464

This theorem is used by:  onexlimgt  44030