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

Theorem dfac12k 10144
Description: Equivalence of dfac12 10146 and dfac12a 10145, without using Regularity. (Contributed by Mario Carneiro, 21-May-2015.)
Assertion
Ref Expression
dfac12k (βˆ€π‘₯ ∈ On 𝒫 π‘₯ ∈ dom card ↔ βˆ€π‘¦ ∈ On 𝒫 (β„΅β€˜π‘¦) ∈ dom card)
Distinct variable group:   π‘₯,𝑦

Proof of Theorem dfac12k
StepHypRef Expression
1 alephon 10066 . . . 4 (β„΅β€˜π‘¦) ∈ On
2 pweq 4615 . . . . . 6 (π‘₯ = (β„΅β€˜π‘¦) β†’ 𝒫 π‘₯ = 𝒫 (β„΅β€˜π‘¦))
32eleq1d 2816 . . . . 5 (π‘₯ = (β„΅β€˜π‘¦) β†’ (𝒫 π‘₯ ∈ dom card ↔ 𝒫 (β„΅β€˜π‘¦) ∈ dom card))
43rspcv 3607 . . . 4 ((β„΅β€˜π‘¦) ∈ On β†’ (βˆ€π‘₯ ∈ On 𝒫 π‘₯ ∈ dom card β†’ 𝒫 (β„΅β€˜π‘¦) ∈ dom card))
51, 4ax-mp 5 . . 3 (βˆ€π‘₯ ∈ On 𝒫 π‘₯ ∈ dom card β†’ 𝒫 (β„΅β€˜π‘¦) ∈ dom card)
65ralrimivw 3148 . 2 (βˆ€π‘₯ ∈ On 𝒫 π‘₯ ∈ dom card β†’ βˆ€π‘¦ ∈ On 𝒫 (β„΅β€˜π‘¦) ∈ dom card)
7 omelon 9643 . . . . . . 7 Ο‰ ∈ On
8 cardon 9941 . . . . . . 7 (cardβ€˜π‘₯) ∈ On
9 ontri1 6397 . . . . . . 7 ((Ο‰ ∈ On ∧ (cardβ€˜π‘₯) ∈ On) β†’ (Ο‰ βŠ† (cardβ€˜π‘₯) ↔ Β¬ (cardβ€˜π‘₯) ∈ Ο‰))
107, 8, 9mp2an 688 . . . . . 6 (Ο‰ βŠ† (cardβ€˜π‘₯) ↔ Β¬ (cardβ€˜π‘₯) ∈ Ο‰)
11 cardidm 9956 . . . . . . . 8 (cardβ€˜(cardβ€˜π‘₯)) = (cardβ€˜π‘₯)
12 cardalephex 10087 . . . . . . . 8 (Ο‰ βŠ† (cardβ€˜π‘₯) β†’ ((cardβ€˜(cardβ€˜π‘₯)) = (cardβ€˜π‘₯) ↔ βˆƒπ‘¦ ∈ On (cardβ€˜π‘₯) = (β„΅β€˜π‘¦)))
1311, 12mpbii 232 . . . . . . 7 (Ο‰ βŠ† (cardβ€˜π‘₯) β†’ βˆƒπ‘¦ ∈ On (cardβ€˜π‘₯) = (β„΅β€˜π‘¦))
14 r19.29 3112 . . . . . . . . 9 ((βˆ€π‘¦ ∈ On 𝒫 (β„΅β€˜π‘¦) ∈ dom card ∧ βˆƒπ‘¦ ∈ On (cardβ€˜π‘₯) = (β„΅β€˜π‘¦)) β†’ βˆƒπ‘¦ ∈ On (𝒫 (β„΅β€˜π‘¦) ∈ dom card ∧ (cardβ€˜π‘₯) = (β„΅β€˜π‘¦)))
15 pweq 4615 . . . . . . . . . . . 12 ((cardβ€˜π‘₯) = (β„΅β€˜π‘¦) β†’ 𝒫 (cardβ€˜π‘₯) = 𝒫 (β„΅β€˜π‘¦))
1615eleq1d 2816 . . . . . . . . . . 11 ((cardβ€˜π‘₯) = (β„΅β€˜π‘¦) β†’ (𝒫 (cardβ€˜π‘₯) ∈ dom card ↔ 𝒫 (β„΅β€˜π‘¦) ∈ dom card))
1716biimparc 478 . . . . . . . . . 10 ((𝒫 (β„΅β€˜π‘¦) ∈ dom card ∧ (cardβ€˜π‘₯) = (β„΅β€˜π‘¦)) β†’ 𝒫 (cardβ€˜π‘₯) ∈ dom card)
1817rexlimivw 3149 . . . . . . . . 9 (βˆƒπ‘¦ ∈ On (𝒫 (β„΅β€˜π‘¦) ∈ dom card ∧ (cardβ€˜π‘₯) = (β„΅β€˜π‘¦)) β†’ 𝒫 (cardβ€˜π‘₯) ∈ dom card)
1914, 18syl 17 . . . . . . . 8 ((βˆ€π‘¦ ∈ On 𝒫 (β„΅β€˜π‘¦) ∈ dom card ∧ βˆƒπ‘¦ ∈ On (cardβ€˜π‘₯) = (β„΅β€˜π‘¦)) β†’ 𝒫 (cardβ€˜π‘₯) ∈ dom card)
2019ex 411 . . . . . . 7 (βˆ€π‘¦ ∈ On 𝒫 (β„΅β€˜π‘¦) ∈ dom card β†’ (βˆƒπ‘¦ ∈ On (cardβ€˜π‘₯) = (β„΅β€˜π‘¦) β†’ 𝒫 (cardβ€˜π‘₯) ∈ dom card))
2113, 20syl5 34 . . . . . 6 (βˆ€π‘¦ ∈ On 𝒫 (β„΅β€˜π‘¦) ∈ dom card β†’ (Ο‰ βŠ† (cardβ€˜π‘₯) β†’ 𝒫 (cardβ€˜π‘₯) ∈ dom card))
2210, 21biimtrrid 242 . . . . 5 (βˆ€π‘¦ ∈ On 𝒫 (β„΅β€˜π‘¦) ∈ dom card β†’ (Β¬ (cardβ€˜π‘₯) ∈ Ο‰ β†’ 𝒫 (cardβ€˜π‘₯) ∈ dom card))
23 nnfi 9169 . . . . . . 7 ((cardβ€˜π‘₯) ∈ Ο‰ β†’ (cardβ€˜π‘₯) ∈ Fin)
24 pwfi 9180 . . . . . . 7 ((cardβ€˜π‘₯) ∈ Fin ↔ 𝒫 (cardβ€˜π‘₯) ∈ Fin)
2523, 24sylib 217 . . . . . 6 ((cardβ€˜π‘₯) ∈ Ο‰ β†’ 𝒫 (cardβ€˜π‘₯) ∈ Fin)
26 finnum 9945 . . . . . 6 (𝒫 (cardβ€˜π‘₯) ∈ Fin β†’ 𝒫 (cardβ€˜π‘₯) ∈ dom card)
2725, 26syl 17 . . . . 5 ((cardβ€˜π‘₯) ∈ Ο‰ β†’ 𝒫 (cardβ€˜π‘₯) ∈ dom card)
2822, 27pm2.61d2 181 . . . 4 (βˆ€π‘¦ ∈ On 𝒫 (β„΅β€˜π‘¦) ∈ dom card β†’ 𝒫 (cardβ€˜π‘₯) ∈ dom card)
29 oncardid 9953 . . . . 5 (π‘₯ ∈ On β†’ (cardβ€˜π‘₯) β‰ˆ π‘₯)
30 pwen 9152 . . . . 5 ((cardβ€˜π‘₯) β‰ˆ π‘₯ β†’ 𝒫 (cardβ€˜π‘₯) β‰ˆ 𝒫 π‘₯)
31 ennum 9944 . . . . 5 (𝒫 (cardβ€˜π‘₯) β‰ˆ 𝒫 π‘₯ β†’ (𝒫 (cardβ€˜π‘₯) ∈ dom card ↔ 𝒫 π‘₯ ∈ dom card))
3229, 30, 313syl 18 . . . 4 (π‘₯ ∈ On β†’ (𝒫 (cardβ€˜π‘₯) ∈ dom card ↔ 𝒫 π‘₯ ∈ dom card))
3328, 32syl5ibcom 244 . . 3 (βˆ€π‘¦ ∈ On 𝒫 (β„΅β€˜π‘¦) ∈ dom card β†’ (π‘₯ ∈ On β†’ 𝒫 π‘₯ ∈ dom card))
3433ralrimiv 3143 . 2 (βˆ€π‘¦ ∈ On 𝒫 (β„΅β€˜π‘¦) ∈ dom card β†’ βˆ€π‘₯ ∈ On 𝒫 π‘₯ ∈ dom card)
356, 34impbii 208 1 (βˆ€π‘₯ ∈ On 𝒫 π‘₯ ∈ dom card ↔ βˆ€π‘¦ ∈ On 𝒫 (β„΅β€˜π‘¦) ∈ dom card)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059  βˆƒwrex 3068   βŠ† wss 3947  π’« cpw 4601   class class class wbr 5147  dom cdm 5675  Oncon0 6363  β€˜cfv 6542  Ο‰com 7857   β‰ˆ cen 8938  Fincfn 8941  cardccrd 9932  β„΅cale 9933
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-inf2 9638
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-2o 8469  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-oi 9507  df-har 9554  df-card 9936  df-aleph 9937
This theorem is referenced by:  dfac12  10146
  Copyright terms: Public domain W3C validator