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Theorem dfac12k 10188
Description: Equivalence of dfac12 10190 and dfac12a 10189, without using Regularity. (Contributed by Mario Carneiro, 21-May-2015.)
Assertion
Ref Expression
dfac12k (∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card ↔ ∀𝑦 ∈ On 𝒫 (ℵ‘𝑦) ∈ dom card)
Distinct variable group:   𝑥,𝑦

Proof of Theorem dfac12k
StepHypRef Expression
1 alephon 10109 . . . 4 (ℵ‘𝑦) ∈ On
2 pweq 4614 . . . . . 6 (𝑥 = (ℵ‘𝑦) → 𝒫 𝑥 = 𝒫 (ℵ‘𝑦))
32eleq1d 2826 . . . . 5 (𝑥 = (ℵ‘𝑦) → (𝒫 𝑥 ∈ dom card ↔ 𝒫 (ℵ‘𝑦) ∈ dom card))
43rspcv 3618 . . . 4 ((ℵ‘𝑦) ∈ On → (∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card → 𝒫 (ℵ‘𝑦) ∈ dom card))
51, 4ax-mp 5 . . 3 (∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card → 𝒫 (ℵ‘𝑦) ∈ dom card)
65ralrimivw 3150 . 2 (∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card → ∀𝑦 ∈ On 𝒫 (ℵ‘𝑦) ∈ dom card)
7 omelon 9686 . . . . . . 7 ω ∈ On
8 cardon 9984 . . . . . . 7 (card‘𝑥) ∈ On
9 ontri1 6418 . . . . . . 7 ((ω ∈ On ∧ (card‘𝑥) ∈ On) → (ω ⊆ (card‘𝑥) ↔ ¬ (card‘𝑥) ∈ ω))
107, 8, 9mp2an 692 . . . . . 6 (ω ⊆ (card‘𝑥) ↔ ¬ (card‘𝑥) ∈ ω)
11 cardidm 9999 . . . . . . . 8 (card‘(card‘𝑥)) = (card‘𝑥)
12 cardalephex 10130 . . . . . . . 8 (ω ⊆ (card‘𝑥) → ((card‘(card‘𝑥)) = (card‘𝑥) ↔ ∃𝑦 ∈ On (card‘𝑥) = (ℵ‘𝑦)))
1311, 12mpbii 233 . . . . . . 7 (ω ⊆ (card‘𝑥) → ∃𝑦 ∈ On (card‘𝑥) = (ℵ‘𝑦))
14 r19.29 3114 . . . . . . . . 9 ((∀𝑦 ∈ On 𝒫 (ℵ‘𝑦) ∈ dom card ∧ ∃𝑦 ∈ On (card‘𝑥) = (ℵ‘𝑦)) → ∃𝑦 ∈ On (𝒫 (ℵ‘𝑦) ∈ dom card ∧ (card‘𝑥) = (ℵ‘𝑦)))
15 pweq 4614 . . . . . . . . . . . 12 ((card‘𝑥) = (ℵ‘𝑦) → 𝒫 (card‘𝑥) = 𝒫 (ℵ‘𝑦))
1615eleq1d 2826 . . . . . . . . . . 11 ((card‘𝑥) = (ℵ‘𝑦) → (𝒫 (card‘𝑥) ∈ dom card ↔ 𝒫 (ℵ‘𝑦) ∈ dom card))
1716biimparc 479 . . . . . . . . . 10 ((𝒫 (ℵ‘𝑦) ∈ dom card ∧ (card‘𝑥) = (ℵ‘𝑦)) → 𝒫 (card‘𝑥) ∈ dom card)
1817rexlimivw 3151 . . . . . . . . 9 (∃𝑦 ∈ On (𝒫 (ℵ‘𝑦) ∈ dom card ∧ (card‘𝑥) = (ℵ‘𝑦)) → 𝒫 (card‘𝑥) ∈ dom card)
1914, 18syl 17 . . . . . . . 8 ((∀𝑦 ∈ On 𝒫 (ℵ‘𝑦) ∈ dom card ∧ ∃𝑦 ∈ On (card‘𝑥) = (ℵ‘𝑦)) → 𝒫 (card‘𝑥) ∈ dom card)
2019ex 412 . . . . . . 7 (∀𝑦 ∈ On 𝒫 (ℵ‘𝑦) ∈ dom card → (∃𝑦 ∈ On (card‘𝑥) = (ℵ‘𝑦) → 𝒫 (card‘𝑥) ∈ dom card))
2113, 20syl5 34 . . . . . 6 (∀𝑦 ∈ On 𝒫 (ℵ‘𝑦) ∈ dom card → (ω ⊆ (card‘𝑥) → 𝒫 (card‘𝑥) ∈ dom card))
2210, 21biimtrrid 243 . . . . 5 (∀𝑦 ∈ On 𝒫 (ℵ‘𝑦) ∈ dom card → (¬ (card‘𝑥) ∈ ω → 𝒫 (card‘𝑥) ∈ dom card))
23 nnfi 9207 . . . . . . 7 ((card‘𝑥) ∈ ω → (card‘𝑥) ∈ Fin)
24 pwfi 9357 . . . . . . 7 ((card‘𝑥) ∈ Fin ↔ 𝒫 (card‘𝑥) ∈ Fin)
2523, 24sylib 218 . . . . . 6 ((card‘𝑥) ∈ ω → 𝒫 (card‘𝑥) ∈ Fin)
26 finnum 9988 . . . . . 6 (𝒫 (card‘𝑥) ∈ Fin → 𝒫 (card‘𝑥) ∈ dom card)
2725, 26syl 17 . . . . 5 ((card‘𝑥) ∈ ω → 𝒫 (card‘𝑥) ∈ dom card)
2822, 27pm2.61d2 181 . . . 4 (∀𝑦 ∈ On 𝒫 (ℵ‘𝑦) ∈ dom card → 𝒫 (card‘𝑥) ∈ dom card)
29 oncardid 9996 . . . . 5 (𝑥 ∈ On → (card‘𝑥) ≈ 𝑥)
30 pwen 9190 . . . . 5 ((card‘𝑥) ≈ 𝑥 → 𝒫 (card‘𝑥) ≈ 𝒫 𝑥)
31 ennum 9987 . . . . 5 (𝒫 (card‘𝑥) ≈ 𝒫 𝑥 → (𝒫 (card‘𝑥) ∈ dom card ↔ 𝒫 𝑥 ∈ dom card))
3229, 30, 313syl 18 . . . 4 (𝑥 ∈ On → (𝒫 (card‘𝑥) ∈ dom card ↔ 𝒫 𝑥 ∈ dom card))
3328, 32syl5ibcom 245 . . 3 (∀𝑦 ∈ On 𝒫 (ℵ‘𝑦) ∈ dom card → (𝑥 ∈ On → 𝒫 𝑥 ∈ dom card))
3433ralrimiv 3145 . 2 (∀𝑦 ∈ On 𝒫 (ℵ‘𝑦) ∈ dom card → ∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card)
356, 34impbii 209 1 (∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card ↔ ∀𝑦 ∈ On 𝒫 (ℵ‘𝑦) ∈ dom card)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  wrex 3070  wss 3951  𝒫 cpw 4600   class class class wbr 5143  dom cdm 5685  Oncon0 6384  cfv 6561  ωcom 7887  cen 8982  Fincfn 8985  cardccrd 9975  cale 9976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-oi 9550  df-har 9597  df-card 9979  df-aleph 9980
This theorem is referenced by:  dfac12  10190
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