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Theorem minregex2 41814
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 41813 . 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 4087 . . . . . . . . . . 11 (𝐴 ∈ (ran card βˆ– Ο‰) β†’ 𝐴 ∈ ran card)
3 iscard4 41812 . . . . . . . . . . 11 ((cardβ€˜π΄) = 𝐴 ↔ 𝐴 ∈ ran card)
42, 3sylibr 233 . . . . . . . . . 10 (𝐴 ∈ (ran card βˆ– Ο‰) β†’ (cardβ€˜π΄) = 𝐴)
54adantr 482 . . . . . . . . 9 ((𝐴 ∈ (ran card βˆ– Ο‰) ∧ 𝑦 ∈ On) β†’ (cardβ€˜π΄) = 𝐴)
6 alephcard 10007 . . . . . . . . . 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 3978 . . . . . . . 8 ((𝐴 ∈ (ran card βˆ– Ο‰) ∧ 𝑦 ∈ On) β†’ ((cardβ€˜π΄) βŠ† (cardβ€˜(β„΅β€˜π‘¦)) ↔ 𝐴 βŠ† (β„΅β€˜π‘¦)))
9 numth3 10407 . . . . . . . . 9 (𝐴 ∈ (ran card βˆ– Ο‰) β†’ 𝐴 ∈ dom card)
10 alephon 10006 . . . . . . . . . 10 (β„΅β€˜π‘¦) ∈ On
11 onenon 9886 . . . . . . . . . 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 9914 . . . . . . . . 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 597 . . . . . . . 8 ((𝐴 ∈ (ran card βˆ– Ο‰) ∧ 𝑦 ∈ On) β†’ ((cardβ€˜π΄) βŠ† (cardβ€˜(β„΅β€˜π‘¦)) ↔ 𝐴 β‰Ό (β„΅β€˜π‘¦)))
158, 14bitr3d 281 . . . . . . 7 ((𝐴 ∈ (ran card βˆ– Ο‰) ∧ 𝑦 ∈ On) β†’ (𝐴 βŠ† (β„΅β€˜π‘¦) ↔ 𝐴 β‰Ό (β„΅β€˜π‘¦)))
16153anbi2d 1442 . . . . . 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 3415 . . . . 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 4913 . . . 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 2748 . . 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 3176 . 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3074  {crab 3408   βˆ– cdif 3908   βŠ† wss 3911  βˆ…c0 4283  βˆ© cint 4908   class class class wbr 5106  dom cdm 5634  ran crn 5635  Oncon0 6318  β€˜cfv 6497  Ο‰com 7803   β‰Ό cdom 8882  cardccrd 9872  β„΅cale 9873  cfccf 9874
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 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-inf2 9578  ax-ac2 10400
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3354  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-se 5590  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-isom 6506  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-1o 8413  df-er 8649  df-map 8768  df-en 8885  df-dom 8886  df-sdom 8887  df-fin 8888  df-oi 9447  df-har 9494  df-card 9876  df-aleph 9877  df-cf 9878  df-acn 9879  df-ac 10053
This theorem is referenced by: (None)
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