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Theorem minregex2 43030
Description: Given any cardinal number 𝐴, there exists an argument π‘₯, which yields the least regular uncountable value of β„΅ which dominates 𝐴. This proof uses AC. (Contributed by RP, 24-Nov-2023.)
Assertion
Ref Expression
minregex2 (𝐴 ∈ (ran card βˆ– Ο‰) β†’ βˆƒπ‘₯ ∈ On π‘₯ = ∩ {𝑦 ∈ On ∣ (βˆ… ∈ 𝑦 ∧ 𝐴 β‰Ό (β„΅β€˜π‘¦) ∧ (cfβ€˜(β„΅β€˜π‘¦)) = (β„΅β€˜π‘¦))})
Distinct variable group:   π‘₯,𝐴,𝑦

Proof of Theorem minregex2
StepHypRef Expression
1 minregex 43029 . 2 (𝐴 ∈ (ran card βˆ– Ο‰) β†’ βˆƒπ‘₯ ∈ On π‘₯ = ∩ {𝑦 ∈ On ∣ (βˆ… ∈ 𝑦 ∧ 𝐴 βŠ† (β„΅β€˜π‘¦) ∧ (cfβ€˜(β„΅β€˜π‘¦)) = (β„΅β€˜π‘¦))})
2 eldifi 4119 . . . . . . . . . . 11 (𝐴 ∈ (ran card βˆ– Ο‰) β†’ 𝐴 ∈ ran card)
3 iscard4 43028 . . . . . . . . . . 11 ((cardβ€˜π΄) = 𝐴 ↔ 𝐴 ∈ ran card)
42, 3sylibr 233 . . . . . . . . . 10 (𝐴 ∈ (ran card βˆ– Ο‰) β†’ (cardβ€˜π΄) = 𝐴)
54adantr 479 . . . . . . . . 9 ((𝐴 ∈ (ran card βˆ– Ο‰) ∧ 𝑦 ∈ On) β†’ (cardβ€˜π΄) = 𝐴)
6 alephcard 10093 . . . . . . . . . 10 (cardβ€˜(β„΅β€˜π‘¦)) = (β„΅β€˜π‘¦)
76a1i 11 . . . . . . . . 9 ((𝐴 ∈ (ran card βˆ– Ο‰) ∧ 𝑦 ∈ On) β†’ (cardβ€˜(β„΅β€˜π‘¦)) = (β„΅β€˜π‘¦))
85, 7sseq12d 4006 . . . . . . . 8 ((𝐴 ∈ (ran card βˆ– Ο‰) ∧ 𝑦 ∈ On) β†’ ((cardβ€˜π΄) βŠ† (cardβ€˜(β„΅β€˜π‘¦)) ↔ 𝐴 βŠ† (β„΅β€˜π‘¦)))
9 numth3 10493 . . . . . . . . 9 (𝐴 ∈ (ran card βˆ– Ο‰) β†’ 𝐴 ∈ dom card)
10 alephon 10092 . . . . . . . . . 10 (β„΅β€˜π‘¦) ∈ On
11 onenon 9972 . . . . . . . . . 10 ((β„΅β€˜π‘¦) ∈ On β†’ (β„΅β€˜π‘¦) ∈ dom card)
1210, 11mp1i 13 . . . . . . . . 9 (𝑦 ∈ On β†’ (β„΅β€˜π‘¦) ∈ dom card)
13 carddom2 10000 . . . . . . . . 9 ((𝐴 ∈ dom card ∧ (β„΅β€˜π‘¦) ∈ dom card) β†’ ((cardβ€˜π΄) βŠ† (cardβ€˜(β„΅β€˜π‘¦)) ↔ 𝐴 β‰Ό (β„΅β€˜π‘¦)))
149, 12, 13syl2an 594 . . . . . . . 8 ((𝐴 ∈ (ran card βˆ– Ο‰) ∧ 𝑦 ∈ On) β†’ ((cardβ€˜π΄) βŠ† (cardβ€˜(β„΅β€˜π‘¦)) ↔ 𝐴 β‰Ό (β„΅β€˜π‘¦)))
158, 14bitr3d 280 . . . . . . 7 ((𝐴 ∈ (ran card βˆ– Ο‰) ∧ 𝑦 ∈ On) β†’ (𝐴 βŠ† (β„΅β€˜π‘¦) ↔ 𝐴 β‰Ό (β„΅β€˜π‘¦)))
16153anbi2d 1437 . . . . . 6 ((𝐴 ∈ (ran card βˆ– Ο‰) ∧ 𝑦 ∈ On) β†’ ((βˆ… ∈ 𝑦 ∧ 𝐴 βŠ† (β„΅β€˜π‘¦) ∧ (cfβ€˜(β„΅β€˜π‘¦)) = (β„΅β€˜π‘¦)) ↔ (βˆ… ∈ 𝑦 ∧ 𝐴 β‰Ό (β„΅β€˜π‘¦) ∧ (cfβ€˜(β„΅β€˜π‘¦)) = (β„΅β€˜π‘¦))))
1716rabbidva 3426 . . . . 5 (𝐴 ∈ (ran card βˆ– Ο‰) β†’ {𝑦 ∈ On ∣ (βˆ… ∈ 𝑦 ∧ 𝐴 βŠ† (β„΅β€˜π‘¦) ∧ (cfβ€˜(β„΅β€˜π‘¦)) = (β„΅β€˜π‘¦))} = {𝑦 ∈ On ∣ (βˆ… ∈ 𝑦 ∧ 𝐴 β‰Ό (β„΅β€˜π‘¦) ∧ (cfβ€˜(β„΅β€˜π‘¦)) = (β„΅β€˜π‘¦))})
1817inteqd 4949 . . . 4 (𝐴 ∈ (ran card βˆ– Ο‰) β†’ ∩ {𝑦 ∈ On ∣ (βˆ… ∈ 𝑦 ∧ 𝐴 βŠ† (β„΅β€˜π‘¦) ∧ (cfβ€˜(β„΅β€˜π‘¦)) = (β„΅β€˜π‘¦))} = ∩ {𝑦 ∈ On ∣ (βˆ… ∈ 𝑦 ∧ 𝐴 β‰Ό (β„΅β€˜π‘¦) ∧ (cfβ€˜(β„΅β€˜π‘¦)) = (β„΅β€˜π‘¦))})
1918eqeq2d 2736 . . 3 (𝐴 ∈ (ran card βˆ– Ο‰) β†’ (π‘₯ = ∩ {𝑦 ∈ On ∣ (βˆ… ∈ 𝑦 ∧ 𝐴 βŠ† (β„΅β€˜π‘¦) ∧ (cfβ€˜(β„΅β€˜π‘¦)) = (β„΅β€˜π‘¦))} ↔ π‘₯ = ∩ {𝑦 ∈ On ∣ (βˆ… ∈ 𝑦 ∧ 𝐴 β‰Ό (β„΅β€˜π‘¦) ∧ (cfβ€˜(β„΅β€˜π‘¦)) = (β„΅β€˜π‘¦))}))
2019rexbidv 3169 . 2 (𝐴 ∈ (ran card βˆ– Ο‰) β†’ (βˆƒπ‘₯ ∈ On π‘₯ = ∩ {𝑦 ∈ On ∣ (βˆ… ∈ 𝑦 ∧ 𝐴 βŠ† (β„΅β€˜π‘¦) ∧ (cfβ€˜(β„΅β€˜π‘¦)) = (β„΅β€˜π‘¦))} ↔ βˆƒπ‘₯ ∈ On π‘₯ = ∩ {𝑦 ∈ On ∣ (βˆ… ∈ 𝑦 ∧ 𝐴 β‰Ό (β„΅β€˜π‘¦) ∧ (cfβ€˜(β„΅β€˜π‘¦)) = (β„΅β€˜π‘¦))}))
211, 20mpbid 231 1 (𝐴 ∈ (ran card βˆ– Ο‰) β†’ βˆƒπ‘₯ ∈ On π‘₯ = ∩ {𝑦 ∈ On ∣ (βˆ… ∈ 𝑦 ∧ 𝐴 β‰Ό (β„΅β€˜π‘¦) ∧ (cfβ€˜(β„΅β€˜π‘¦)) = (β„΅β€˜π‘¦))})
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆƒwrex 3060  {crab 3419   βˆ– cdif 3936   βŠ† wss 3939  βˆ…c0 4318  βˆ© cint 4944   class class class wbr 5143  dom cdm 5672  ran crn 5673  Oncon0 6364  β€˜cfv 6543  Ο‰com 7868   β‰Ό cdom 8960  cardccrd 9958  β„΅cale 9959  cfccf 9960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738  ax-inf2 9664  ax-ac2 10486
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-1st 7991  df-2nd 7992  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-er 8723  df-map 8845  df-en 8963  df-dom 8964  df-sdom 8965  df-fin 8966  df-oi 9533  df-har 9580  df-card 9962  df-aleph 9963  df-cf 9964  df-acn 9965  df-ac 10139
This theorem is referenced by: (None)
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