Theorem minregex2 42286

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 42285	. 2 ⊢ (𝐴 ∈ (ran card ∖ ω) → ∃𝑥 ∈ On 𝑥 = ∩ {𝑦 ∈ On ∣ (∅ ∈ 𝑦 ∧ 𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))})
2		eldifi 4127	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (ran card ∖ ω) → 𝐴 ∈ ran card)
3		iscard4 42284	. . . . . . . . . . 11 ⊢ ((card‘𝐴) = 𝐴 ↔ 𝐴 ∈ ran card)
4	2, 3	sylibr 233	. . . . . . . . . 10 ⊢ (𝐴 ∈ (ran card ∖ ω) → (card‘𝐴) = 𝐴)
5	4	adantr 482	. . . . . . . . 9 ⊢ ((𝐴 ∈ (ran card ∖ ω) ∧ 𝑦 ∈ On) → (card‘𝐴) = 𝐴)
6		alephcard 10065	. . . . . . . . . 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 4016	. . . . . . . 8 ⊢ ((𝐴 ∈ (ran card ∖ ω) ∧ 𝑦 ∈ On) → ((card‘𝐴) ⊆ (card‘(ℵ‘𝑦)) ↔ 𝐴 ⊆ (ℵ‘𝑦)))
9		numth3 10465	. . . . . . . . 9 ⊢ (𝐴 ∈ (ran card ∖ ω) → 𝐴 ∈ dom card)
10		alephon 10064	. . . . . . . . . 10 ⊢ (ℵ‘𝑦) ∈ On
11		onenon 9944	. . . . . . . . . 10 ⊢ ((ℵ‘𝑦) ∈ On → (ℵ‘𝑦) ∈ dom card)
12	10, 11	mp1i 13	. . . . . . . . 9 ⊢ (𝑦 ∈ On → (ℵ‘𝑦) ∈ dom card)
13		carddom2 9972	. . . . . . . . 9 ⊢ ((𝐴 ∈ dom card ∧ (ℵ‘𝑦) ∈ dom card) → ((card‘𝐴) ⊆ (card‘(ℵ‘𝑦)) ↔ 𝐴 ≼ (ℵ‘𝑦)))
14	9, 12, 13	syl2an 597	. . . . . . . 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 1442	. . . . . 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 3440	. . . . 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 4956	. . . 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 2744	. . 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 3179	. 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 231	1 ⊢ (𝐴 ∈ (ran card ∖ ω) → ∃𝑥 ∈ On 𝑥 = ∩ {𝑦 ∈ On ∣ (∅ ∈ 𝑦 ∧ 𝐴 ≼ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))})

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  ∃wrex 3071  {crab 3433   ∖ cdif 3946   ⊆ wss 3949  ∅c0 4323  ∩ cint 4951   class class class wbr 5149  dom cdm 5677  ran crn 5678  Oncon0 6365  ‘cfv 6544  ωcom 7855   ≼ cdom 8937  cardccrd 9930  ℵcale 9931  cfccf 9932

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-inf2 9636  ax-ac2 10458

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-map 8822  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-oi 9505  df-har 9552  df-card 9934  df-aleph 9935  df-cf 9936  df-acn 9937  df-ac 10111

This theorem is referenced by: (None)