MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfac12k Structured version   Visualization version   GIF version

Theorem dfac12k 9562
Description: Equivalence of dfac12 9564 and dfac12a 9563, 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 9484 . . . 4 (ℵ‘𝑦) ∈ On
2 pweq 4516 . . . . . 6 (𝑥 = (ℵ‘𝑦) → 𝒫 𝑥 = 𝒫 (ℵ‘𝑦))
32eleq1d 2877 . . . . 5 (𝑥 = (ℵ‘𝑦) → (𝒫 𝑥 ∈ dom card ↔ 𝒫 (ℵ‘𝑦) ∈ dom card))
43rspcv 3569 . . . 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 3153 . 2 (∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card → ∀𝑦 ∈ On 𝒫 (ℵ‘𝑦) ∈ dom card)
7 omelon 9097 . . . . . . 7 ω ∈ On
8 cardon 9361 . . . . . . 7 (card‘𝑥) ∈ On
9 ontri1 6197 . . . . . . 7 ((ω ∈ On ∧ (card‘𝑥) ∈ On) → (ω ⊆ (card‘𝑥) ↔ ¬ (card‘𝑥) ∈ ω))
107, 8, 9mp2an 691 . . . . . 6 (ω ⊆ (card‘𝑥) ↔ ¬ (card‘𝑥) ∈ ω)
11 cardidm 9376 . . . . . . . 8 (card‘(card‘𝑥)) = (card‘𝑥)
12 cardalephex 9505 . . . . . . . 8 (ω ⊆ (card‘𝑥) → ((card‘(card‘𝑥)) = (card‘𝑥) ↔ ∃𝑦 ∈ On (card‘𝑥) = (ℵ‘𝑦)))
1311, 12mpbii 236 . . . . . . 7 (ω ⊆ (card‘𝑥) → ∃𝑦 ∈ On (card‘𝑥) = (ℵ‘𝑦))
14 r19.29 3219 . . . . . . . . 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 4516 . . . . . . . . . . . 12 ((card‘𝑥) = (ℵ‘𝑦) → 𝒫 (card‘𝑥) = 𝒫 (ℵ‘𝑦))
1615eleq1d 2877 . . . . . . . . . . 11 ((card‘𝑥) = (ℵ‘𝑦) → (𝒫 (card‘𝑥) ∈ dom card ↔ 𝒫 (ℵ‘𝑦) ∈ dom card))
1716biimparc 483 . . . . . . . . . 10 ((𝒫 (ℵ‘𝑦) ∈ dom card ∧ (card‘𝑥) = (ℵ‘𝑦)) → 𝒫 (card‘𝑥) ∈ dom card)
1817rexlimivw 3244 . . . . . . . . 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 416 . . . . . . 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, 21syl5bir 246 . . . . 5 (∀𝑦 ∈ On 𝒫 (ℵ‘𝑦) ∈ dom card → (¬ (card‘𝑥) ∈ ω → 𝒫 (card‘𝑥) ∈ dom card))
23 nnfi 8700 . . . . . . 7 ((card‘𝑥) ∈ ω → (card‘𝑥) ∈ Fin)
24 pwfi 8807 . . . . . . 7 ((card‘𝑥) ∈ Fin ↔ 𝒫 (card‘𝑥) ∈ Fin)
2523, 24sylib 221 . . . . . 6 ((card‘𝑥) ∈ ω → 𝒫 (card‘𝑥) ∈ Fin)
26 finnum 9365 . . . . . 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 184 . . . 4 (∀𝑦 ∈ On 𝒫 (ℵ‘𝑦) ∈ dom card → 𝒫 (card‘𝑥) ∈ dom card)
29 oncardid 9373 . . . . 5 (𝑥 ∈ On → (card‘𝑥) ≈ 𝑥)
30 pwen 8678 . . . . 5 ((card‘𝑥) ≈ 𝑥 → 𝒫 (card‘𝑥) ≈ 𝒫 𝑥)
31 ennum 9364 . . . . 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 248 . . 3 (∀𝑦 ∈ On 𝒫 (ℵ‘𝑦) ∈ dom card → (𝑥 ∈ On → 𝒫 𝑥 ∈ dom card))
3433ralrimiv 3151 . 2 (∀𝑦 ∈ On 𝒫 (ℵ‘𝑦) ∈ dom card → ∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card)
356, 34impbii 212 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 209  wa 399   = wceq 1538  wcel 2112  wral 3109  wrex 3110  wss 3884  𝒫 cpw 4500   class class class wbr 5033  dom cdm 5523  Oncon0 6163  cfv 6328  ωcom 7564  cen 8493  Fincfn 8496  cardccrd 9352  cale 9353
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445  ax-inf2 9092
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-se 5483  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-isom 6337  df-riota 7097  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-oadd 8093  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-oi 8962  df-har 9009  df-card 9356  df-aleph 9357
This theorem is referenced by:  dfac12  9564
  Copyright terms: Public domain W3C validator