Theorem onexomgt 43818

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 9601	. . 3 ⊢ ω ∈ On
2		peano1 7869	. . . 4 ⊢ ∅ ∈ ω
3	2	ne0ii 4296	. . 3 ⊢ ω ≠ ∅
4		omeu 8554	. . 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 1473	. 2 ⊢ (𝐴 ∈ On → ∃!𝑐∃𝑎 ∈ On ∃𝑏 ∈ ω (𝑐 = ⟨𝑎, 𝑏⟩ ∧ ((ω ·o 𝑎) +o 𝑏) = 𝐴))
6		euex 2604	. . 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 7793	. . . . . . . . . 10 ⊢ (𝑎 ∈ On → suc 𝑎 ∈ On)
8	7	adantr 484	. . . . . . . . 9 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ ω) → suc 𝑎 ∈ On)
9		simpr 488	. . . . . . . . . 10 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ ω) ∧ ((ω ·o 𝑎) +o 𝑏) = 𝐴) → ((ω ·o 𝑎) +o 𝑏) = 𝐴)
10		simpr 488	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ ω) → 𝑏 ∈ ω)
11		simpl 486	. . . . . . . . . . . . . . 15 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ ω) → 𝑎 ∈ On)
12		omcl 8505	. . . . . . . . . . . . . . 15 ⊢ ((ω ∈ On ∧ 𝑎 ∈ On) → (ω ·o 𝑎) ∈ On)
13	1, 11, 12	sylancr 596	. . . . . . . . . . . . . 14 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ ω) → (ω ·o 𝑎) ∈ On)
14		oaordi 8515	. . . . . . . . . . . . . 14 ⊢ ((ω ∈ On ∧ (ω ·o 𝑎) ∈ On) → (𝑏 ∈ ω → ((ω ·o 𝑎) +o 𝑏) ∈ ((ω ·o 𝑎) +o ω)))
15	1, 13, 14	sylancr 596	. . . . . . . . . . . . 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 8495	. . . . . . . . . . . . 13 ⊢ ((ω ∈ On ∧ 𝑎 ∈ On) → (ω ·o suc 𝑎) = ((ω ·o 𝑎) +o ω))
18	1, 11, 17	sylancr 596	. . . . . . . . . . . 12 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ ω) → (ω ·o suc 𝑎) = ((ω ·o 𝑎) +o ω))
19	16, 18	eleqtrrd 2865	. . . . . . . . . . 11 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ ω) → ((ω ·o 𝑎) +o 𝑏) ∈ (ω ·o suc 𝑎))
20	19	adantr 484	. . . . . . . . . 10 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ ω) ∧ ((ω ·o 𝑎) +o 𝑏) = 𝐴) → ((ω ·o 𝑎) +o 𝑏) ∈ (ω ·o suc 𝑎))
21	9, 20	eqeltrrd 2863	. . . . . . . . 9 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ ω) ∧ ((ω ·o 𝑎) +o 𝑏) = 𝐴) → 𝐴 ∈ (ω ·o suc 𝑎))
22		oveq2 7404	. . . . . . . . . . 11 ⊢ (𝑥 = suc 𝑎 → (ω ·o 𝑥) = (ω ·o suc 𝑎))
23	22	eleq2d 2848	. . . . . . . . . 10 ⊢ (𝑥 = suc 𝑎 → (𝐴 ∈ (ω ·o 𝑥) ↔ 𝐴 ∈ (ω ·o suc 𝑎)))
24	23	rspcev 3581	. . . . . . . . 9 ⊢ ((suc 𝑎 ∈ On ∧ 𝐴 ∈ (ω ·o suc 𝑎)) → ∃𝑥 ∈ On 𝐴 ∈ (ω ·o 𝑥))
25	8, 21, 24	syl2an2r 695	. . . . . . . 8 ⊢ (((𝑎 ∈ On ∧ 𝑏 ∈ ω) ∧ ((ω ·o 𝑎) +o 𝑏) = 𝐴) → ∃𝑥 ∈ On 𝐴 ∈ (ω ·o 𝑥))
26	25	ex 416	. . . . . . 7 ⊢ ((𝑎 ∈ On ∧ 𝑏 ∈ ω) → (((ω ·o 𝑎) +o 𝑏) = 𝐴 → ∃𝑥 ∈ On 𝐴 ∈ (ω ·o 𝑥)))
27	26	adantld 494	. . . . . 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 3218	. . . 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 1953	. . 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 399   = wceq 1560  ∃wex 1799   ∈ wcel 2142  ∃!weu 2595   ≠ wne 2957  ∃wrex 3086  ∅c0 4285  ⟨cop 4588  Oncon0 6346  suc csuc 6348  (class class class)co 7396  ωcom 7846   +_o coa 8434   ·_o comu 8435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5227  ax-sep 5246  ax-nul 5256  ax-pr 5390  ax-un 7718  ax-inf2 9596

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3077  df-rex 3087  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3456  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4481  df-pw 4557  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4906  df-iun 4951  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6288  df-ord 6349  df-on 6350  df-lim 6351  df-suc 6352  df-iota 6477  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528  df-fv 6529  df-ov 7399  df-oprab 7400  df-mpo 7401  df-om 7847  df-2nd 7971  df-frecs 8262  df-wrecs 8293  df-recs 8342  df-rdg 8381  df-1o 8437  df-oadd 8441  df-omul 8442

This theorem is referenced by:  onexlimgt  43820