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Theorem minregex2 43497
Description: Given any cardinal number 𝐴, there exists an argument 𝑥, which yields the least regular uncountable value of 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 ∣ (∅ ∈ 𝑦𝐴 ≼ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))})
Distinct variable group:   𝑥,𝐴,𝑦

Proof of Theorem minregex2
StepHypRef Expression
1 minregex 43496 . 2 (𝐴 ∈ (ran card ∖ ω) → ∃𝑥 ∈ On 𝑥 = {𝑦 ∈ On ∣ (∅ ∈ 𝑦𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))})
2 eldifi 4154 . . . . . . . . . . 11 (𝐴 ∈ (ran card ∖ ω) → 𝐴 ∈ ran card)
3 iscard4 43495 . . . . . . . . . . 11 ((card‘𝐴) = 𝐴𝐴 ∈ ran card)
42, 3sylibr 234 . . . . . . . . . 10 (𝐴 ∈ (ran card ∖ ω) → (card‘𝐴) = 𝐴)
54adantr 480 . . . . . . . . 9 ((𝐴 ∈ (ran card ∖ ω) ∧ 𝑦 ∈ On) → (card‘𝐴) = 𝐴)
6 alephcard 10139 . . . . . . . . . 10 (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)
76a1i 11 . . . . . . . . 9 ((𝐴 ∈ (ran card ∖ ω) ∧ 𝑦 ∈ On) → (card‘(ℵ‘𝑦)) = (ℵ‘𝑦))
85, 7sseq12d 4042 . . . . . . . 8 ((𝐴 ∈ (ran card ∖ ω) ∧ 𝑦 ∈ On) → ((card‘𝐴) ⊆ (card‘(ℵ‘𝑦)) ↔ 𝐴 ⊆ (ℵ‘𝑦)))
9 numth3 10539 . . . . . . . . 9 (𝐴 ∈ (ran card ∖ ω) → 𝐴 ∈ dom card)
10 alephon 10138 . . . . . . . . . 10 (ℵ‘𝑦) ∈ On
11 onenon 10018 . . . . . . . . . 10 ((ℵ‘𝑦) ∈ On → (ℵ‘𝑦) ∈ dom card)
1210, 11mp1i 13 . . . . . . . . 9 (𝑦 ∈ On → (ℵ‘𝑦) ∈ dom card)
13 carddom2 10046 . . . . . . . . 9 ((𝐴 ∈ dom card ∧ (ℵ‘𝑦) ∈ dom card) → ((card‘𝐴) ⊆ (card‘(ℵ‘𝑦)) ↔ 𝐴 ≼ (ℵ‘𝑦)))
149, 12, 13syl2an 595 . . . . . . . 8 ((𝐴 ∈ (ran card ∖ ω) ∧ 𝑦 ∈ On) → ((card‘𝐴) ⊆ (card‘(ℵ‘𝑦)) ↔ 𝐴 ≼ (ℵ‘𝑦)))
158, 14bitr3d 281 . . . . . . 7 ((𝐴 ∈ (ran card ∖ ω) ∧ 𝑦 ∈ On) → (𝐴 ⊆ (ℵ‘𝑦) ↔ 𝐴 ≼ (ℵ‘𝑦)))
16153anbi2d 1441 . . . . . 6 ((𝐴 ∈ (ran card ∖ ω) ∧ 𝑦 ∈ On) → ((∅ ∈ 𝑦𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦)) ↔ (∅ ∈ 𝑦𝐴 ≼ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))))
1716rabbidva 3450 . . . . 5 (𝐴 ∈ (ran card ∖ ω) → {𝑦 ∈ On ∣ (∅ ∈ 𝑦𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))} = {𝑦 ∈ On ∣ (∅ ∈ 𝑦𝐴 ≼ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))})
1817inteqd 4975 . . . 4 (𝐴 ∈ (ran card ∖ ω) → {𝑦 ∈ On ∣ (∅ ∈ 𝑦𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))} = {𝑦 ∈ On ∣ (∅ ∈ 𝑦𝐴 ≼ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))})
1918eqeq2d 2751 . . 3 (𝐴 ∈ (ran card ∖ ω) → (𝑥 = {𝑦 ∈ On ∣ (∅ ∈ 𝑦𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))} ↔ 𝑥 = {𝑦 ∈ On ∣ (∅ ∈ 𝑦𝐴 ≼ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))}))
2019rexbidv 3185 . 2 (𝐴 ∈ (ran card ∖ ω) → (∃𝑥 ∈ On 𝑥 = {𝑦 ∈ On ∣ (∅ ∈ 𝑦𝐴 ⊆ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))} ↔ ∃𝑥 ∈ On 𝑥 = {𝑦 ∈ On ∣ (∅ ∈ 𝑦𝐴 ≼ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))}))
211, 20mpbid 232 1 (𝐴 ∈ (ran card ∖ ω) → ∃𝑥 ∈ On 𝑥 = {𝑦 ∈ On ∣ (∅ ∈ 𝑦𝐴 ≼ (ℵ‘𝑦) ∧ (cf‘(ℵ‘𝑦)) = (ℵ‘𝑦))})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1537  wcel 2108  wrex 3076  {crab 3443  cdif 3973  wss 3976  c0 4352   cint 4970   class class class wbr 5166  dom cdm 5700  ran crn 5701  Oncon0 6395  cfv 6573  ωcom 7903  cdom 9001  cardccrd 10004  cale 10005  cfccf 10006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-inf2 9710  ax-ac2 10532
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-oi 9579  df-har 9626  df-card 10008  df-aleph 10009  df-cf 10010  df-acn 10011  df-ac 10185
This theorem is referenced by: (None)
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