Theorem minregex2 44360

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 44359	. 2 ⊢ (𝐴 ∈ (ran card ∖ ω) → ∃𝑥 ∈ On 𝑥 = ∩ {𝑦 ∈ On ∣ (∅ ∈ 𝑦 ∧ 𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))})
2		eldifi 4081	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (ran card ∖ ω) → 𝐴 ∈ ran card)
3		iscard4 44358	. . . . . . . . . . 11 ⊢ ((card‘𝐴) = 𝐴 ↔ 𝐴 ∈ ran card)
4	2, 3	sylibr 237	. . . . . . . . . 10 ⊢ (𝐴 ∈ (ran card ∖ ω) → (card‘𝐴) = 𝐴)
5	4	adantr 486	. . . . . . . . 9 ⊢ ((𝐴 ∈ (ran card ∖ ω) ∧ 𝑦 ∈ On) → (card‘𝐴) = 𝐴)
6		alephcard 10076	. . . . . . . . . 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 3967	. . . . . . . 8 ⊢ ((𝐴 ∈ (ran card ∖ ω) ∧ 𝑦 ∈ On) → ((card‘𝐴) ⊆ (card‘(ℵ‘𝑦)) ↔ 𝐴 ⊆ (ℵ‘𝑦)))
9		numth3 10475	. . . . . . . . 9 ⊢ (𝐴 ∈ (ran card ∖ ω) → 𝐴 ∈ dom card)
10		alephon 10075	. . . . . . . . . 10 ⊢ (ℵ‘𝑦) ∈ On
11		onenon 9957	. . . . . . . . . 10 ⊢ ((ℵ‘𝑦) ∈ On → (ℵ‘𝑦) ∈ dom card)
12	10, 11	mp1i 14	. . . . . . . . 9 ⊢ (𝑦 ∈ On → (ℵ‘𝑦) ∈ dom card)
13		carddom2 9985	. . . . . . . . 9 ⊢ ((𝐴 ∈ dom card ∧ (ℵ‘𝑦) ∈ dom card) → ((card‘𝐴) ⊆ (card‘(ℵ‘𝑦)) ↔ 𝐴 ≼ (ℵ‘𝑦)))
14	9, 12, 13	syl2an 608	. . . . . . . 8 ⊢ ((𝐴 ∈ (ran card ∖ ω) ∧ 𝑦 ∈ On) → ((card‘𝐴) ⊆ (card‘(ℵ‘𝑦)) ↔ 𝐴 ≼ (ℵ‘𝑦)))
15	8, 14	bitr3d 284	. . . . . . 7 ⊢ ((𝐴 ∈ (ran card ∖ ω) ∧ 𝑦 ∈ On) → (𝐴 ⊆ (ℵ‘𝑦) ↔ 𝐴 ≼ (ℵ‘𝑦)))
16	15	3anbi2d 1469	. . . . . 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 3420	. . . . 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 4915	. . . 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 2773	. . 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 3188	. 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 235	1 ⊢ (𝐴 ∈ (ran card ∖ ω) → ∃𝑥 ∈ On 𝑥 = ∩ {𝑦 ∈ On ∣ (∅ ∈ 𝑦 ∧ 𝐴 ≼ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∃wrex 3088  {crab 3414   ∖ cdif 3899   ⊆ wss 3902  ∅c0 4282  ∩ cint 4910   class class class wbr 5107  dom cdm 5659  ran crn 5660  Oncon0 6361  ‘cfv 6537  ωcom 7865   ≼ cdom 8953  cardccrd 9943  ℵcale 9944  cfccf 9945

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739  ax-inf2 9623  ax-ac2 10468

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  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 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7866  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-rdg 8402  df-1o 8458  df-er 8699  df-map 8831  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-oi 9485  df-har 9532  df-card 9947  df-aleph 9948  df-cf 9949  df-acn 9950  df-ac 10122

This theorem is used by: (None)