Theorem minregex2 43534

Description: Given any cardinal number 𝐴, there exists an argument 𝑥, which yields the least regular uncountable value of ℵ which dominates 𝐴. This proof uses AC. (Contributed by RP, 24-Nov-2023.)

Assertion

Ref	Expression
minregex2	⊢ (𝐴 ∈ (ran card ∖ ω) → ∃𝑥 ∈ On 𝑥 = ∩ {𝑦 ∈ On ∣ (∅ ∈ 𝑦 ∧ 𝐴 ≼ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))})

Distinct variable group:   𝑥,𝐴,𝑦

Proof of Theorem minregex2

Step	Hyp	Ref	Expression
1		minregex 43533	. 2 ⊢ (𝐴 ∈ (ran card ∖ ω) → ∃𝑥 ∈ On 𝑥 = ∩ {𝑦 ∈ On ∣ (∅ ∈ 𝑦 ∧ 𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))})
2		eldifi 4111	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (ran card ∖ ω) → 𝐴 ∈ ran card)
3		iscard4 43532	. . . . . . . . . . 11 ⊢ ((card‘𝐴) = 𝐴 ↔ 𝐴 ∈ ran card)
4	2, 3	sylibr 234	. . . . . . . . . 10 ⊢ (𝐴 ∈ (ran card ∖ ω) → (card‘𝐴) = 𝐴)
5	4	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ (ran card ∖ ω) ∧ 𝑦 ∈ On) → (card‘𝐴) = 𝐴)
6		alephcard 10089	. . . . . . . . . 10 ⊢ (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)
7	6	a1i 11	. . . . . . . . 9 ⊢ ((𝐴 ∈ (ran card ∖ ω) ∧ 𝑦 ∈ On) → (card‘(ℵ‘𝑦)) = (ℵ‘𝑦))
8	5, 7	sseq12d 3997	. . . . . . . 8 ⊢ ((𝐴 ∈ (ran card ∖ ω) ∧ 𝑦 ∈ On) → ((card‘𝐴) ⊆ (card‘(ℵ‘𝑦)) ↔ 𝐴 ⊆ (ℵ‘𝑦)))
9		numth3 10489	. . . . . . . . 9 ⊢ (𝐴 ∈ (ran card ∖ ω) → 𝐴 ∈ dom card)
10		alephon 10088	. . . . . . . . . 10 ⊢ (ℵ‘𝑦) ∈ On
11		onenon 9968	. . . . . . . . . 10 ⊢ ((ℵ‘𝑦) ∈ On → (ℵ‘𝑦) ∈ dom card)
12	10, 11	mp1i 13	. . . . . . . . 9 ⊢ (𝑦 ∈ On → (ℵ‘𝑦) ∈ dom card)
13		carddom2 9996	. . . . . . . . 9 ⊢ ((𝐴 ∈ dom card ∧ (ℵ‘𝑦) ∈ dom card) → ((card‘𝐴) ⊆ (card‘(ℵ‘𝑦)) ↔ 𝐴 ≼ (ℵ‘𝑦)))
14	9, 12, 13	syl2an 596	. . . . . . . 8 ⊢ ((𝐴 ∈ (ran card ∖ ω) ∧ 𝑦 ∈ On) → ((card‘𝐴) ⊆ (card‘(ℵ‘𝑦)) ↔ 𝐴 ≼ (ℵ‘𝑦)))
15	8, 14	bitr3d 281	. . . . . . 7 ⊢ ((𝐴 ∈ (ran card ∖ ω) ∧ 𝑦 ∈ On) → (𝐴 ⊆ (ℵ‘𝑦) ↔ 𝐴 ≼ (ℵ‘𝑦)))
16	15	3anbi2d 1443	. . . . . 6 ⊢ ((𝐴 ∈ (ran card ∖ ω) ∧ 𝑦 ∈ On) → ((∅ ∈ 𝑦 ∧ 𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦)) ↔ (∅ ∈ 𝑦 ∧ 𝐴 ≼ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))))
17	16	rabbidva 3427	. . . . 5 ⊢ (𝐴 ∈ (ran card ∖ ω) → {𝑦 ∈ On ∣ (∅ ∈ 𝑦 ∧ 𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))} = {𝑦 ∈ On ∣ (∅ ∈ 𝑦 ∧ 𝐴 ≼ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))})
18	17	inteqd 4932	. . . 4 ⊢ (𝐴 ∈ (ran card ∖ ω) → ∩ {𝑦 ∈ On ∣ (∅ ∈ 𝑦 ∧ 𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))} = ∩ {𝑦 ∈ On ∣ (∅ ∈ 𝑦 ∧ 𝐴 ≼ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))})
19	18	eqeq2d 2747	. . 3 ⊢ (𝐴 ∈ (ran card ∖ ω) → (𝑥 = ∩ {𝑦 ∈ On ∣ (∅ ∈ 𝑦 ∧ 𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))} ↔ 𝑥 = ∩ {𝑦 ∈ On ∣ (∅ ∈ 𝑦 ∧ 𝐴 ≼ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))}))
20	19	rexbidv 3165	. 2 ⊢ (𝐴 ∈ (ran card ∖ ω) → (∃𝑥 ∈ On 𝑥 = ∩ {𝑦 ∈ On ∣ (∅ ∈ 𝑦 ∧ 𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))} ↔ ∃𝑥 ∈ On 𝑥 = ∩ {𝑦 ∈ On ∣ (∅ ∈ 𝑦 ∧ 𝐴 ≼ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))}))
21	1, 20	mpbid 232	1 ⊢ (𝐴 ∈ (ran card ∖ ω) → ∃𝑥 ∈ On 𝑥 = ∩ {𝑦 ∈ On ∣ (∅ ∈ 𝑦 ∧ 𝐴 ≼ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))})

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∃wrex 3061  {crab 3420   ∖ cdif 3928   ⊆ wss 3931  ∅c0 4313  ∩ cint 4927   class class class wbr 5124  dom cdm 5659  ran crn 5660  Oncon0 6357  ‘cfv 6536  ωcom 7866   ≼ cdom 8962  cardccrd 9954  ℵcale 9955  cfccf 9956

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-inf2 9660  ax-ac2 10482

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3364  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-int 4928  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-tr 5235  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-se 5612  df-we 5613  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6295  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-isom 6545  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-er 8724  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-fin 8968  df-oi 9529  df-har 9576  df-card 9958  df-aleph 9959  df-cf 9960  df-acn 9961  df-ac 10135

This theorem is referenced by: (None)