Theorem minregex2 43895

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 43894	. 2 ⊢ (𝐴 ∈ (ran card ∖ ω) → ∃𝑥 ∈ On 𝑥 = ∩ {𝑦 ∈ On ∣ (∅ ∈ 𝑦 ∧ 𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))})
2		eldifi 4085	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (ran card ∖ ω) → 𝐴 ∈ ran card)
3		iscard4 43893	. . . . . . . . . . 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 9992	. . . . . . . . . 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 3969	. . . . . . . 8 ⊢ ((𝐴 ∈ (ran card ∖ ω) ∧ 𝑦 ∈ On) → ((card‘𝐴) ⊆ (card‘(ℵ‘𝑦)) ↔ 𝐴 ⊆ (ℵ‘𝑦)))
9		numth3 10392	. . . . . . . . 9 ⊢ (𝐴 ∈ (ran card ∖ ω) → 𝐴 ∈ dom card)
10		alephon 9991	. . . . . . . . . 10 ⊢ (ℵ‘𝑦) ∈ On
11		onenon 9873	. . . . . . . . . 10 ⊢ ((ℵ‘𝑦) ∈ On → (ℵ‘𝑦) ∈ dom card)
12	10, 11	mp1i 13	. . . . . . . . 9 ⊢ (𝑦 ∈ On → (ℵ‘𝑦) ∈ dom card)
13		carddom2 9901	. . . . . . . . 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 1444	. . . . . 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 3407	. . . . 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 4909	. . . 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 2748	. . 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 3162	. 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 1087   = wceq 1542   ∈ wcel 2114  ∃wrex 3062  {crab 3401   ∖ cdif 3900   ⊆ wss 3903  ∅c0 4287  ∩ cint 4904   class class class wbr 5100  dom cdm 5632  ran crn 5633  Oncon0 6325  ‘cfv 6500  ωcom 7818   ≼ cdom 8893  cardccrd 9859  ℵcale 9860  cfccf 9861

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-inf2 9562  ax-ac2 10385

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-se 5586  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-er 8645  df-map 8777  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-oi 9427  df-har 9474  df-card 9863  df-aleph 9864  df-cf 9865  df-acn 9866  df-ac 10038

This theorem is referenced by: (None)