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

Theorem dfac12k 10091
Description: Equivalence of dfac12 10093 and dfac12a 10092, 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 10013 . . . 4 (β„΅β€˜π‘¦) ∈ On
2 pweq 4578 . . . . . 6 (π‘₯ = (β„΅β€˜π‘¦) β†’ 𝒫 π‘₯ = 𝒫 (β„΅β€˜π‘¦))
32eleq1d 2819 . . . . 5 (π‘₯ = (β„΅β€˜π‘¦) β†’ (𝒫 π‘₯ ∈ dom card ↔ 𝒫 (β„΅β€˜π‘¦) ∈ dom card))
43rspcv 3579 . . . 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 3144 . 2 (βˆ€π‘₯ ∈ On 𝒫 π‘₯ ∈ dom card β†’ βˆ€π‘¦ ∈ On 𝒫 (β„΅β€˜π‘¦) ∈ dom card)
7 omelon 9590 . . . . . . 7 Ο‰ ∈ On
8 cardon 9888 . . . . . . 7 (cardβ€˜π‘₯) ∈ On
9 ontri1 6355 . . . . . . 7 ((Ο‰ ∈ On ∧ (cardβ€˜π‘₯) ∈ On) β†’ (Ο‰ βŠ† (cardβ€˜π‘₯) ↔ Β¬ (cardβ€˜π‘₯) ∈ Ο‰))
107, 8, 9mp2an 691 . . . . . 6 (Ο‰ βŠ† (cardβ€˜π‘₯) ↔ Β¬ (cardβ€˜π‘₯) ∈ Ο‰)
11 cardidm 9903 . . . . . . . 8 (cardβ€˜(cardβ€˜π‘₯)) = (cardβ€˜π‘₯)
12 cardalephex 10034 . . . . . . . 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 3114 . . . . . . . . 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 4578 . . . . . . . . . . . 12 ((cardβ€˜π‘₯) = (β„΅β€˜π‘¦) β†’ 𝒫 (cardβ€˜π‘₯) = 𝒫 (β„΅β€˜π‘¦))
1615eleq1d 2819 . . . . . . . . . . 11 ((cardβ€˜π‘₯) = (β„΅β€˜π‘¦) β†’ (𝒫 (cardβ€˜π‘₯) ∈ dom card ↔ 𝒫 (β„΅β€˜π‘¦) ∈ dom card))
1716biimparc 481 . . . . . . . . . 10 ((𝒫 (β„΅β€˜π‘¦) ∈ dom card ∧ (cardβ€˜π‘₯) = (β„΅β€˜π‘¦)) β†’ 𝒫 (cardβ€˜π‘₯) ∈ dom card)
1817rexlimivw 3145 . . . . . . . . 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 414 . . . . . . 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 9117 . . . . . . 7 ((cardβ€˜π‘₯) ∈ Ο‰ β†’ (cardβ€˜π‘₯) ∈ Fin)
24 pwfi 9128 . . . . . . 7 ((cardβ€˜π‘₯) ∈ Fin ↔ 𝒫 (cardβ€˜π‘₯) ∈ Fin)
2523, 24sylib 217 . . . . . 6 ((cardβ€˜π‘₯) ∈ Ο‰ β†’ 𝒫 (cardβ€˜π‘₯) ∈ Fin)
26 finnum 9892 . . . . . 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 9900 . . . . 5 (π‘₯ ∈ On β†’ (cardβ€˜π‘₯) β‰ˆ π‘₯)
30 pwen 9100 . . . . 5 ((cardβ€˜π‘₯) β‰ˆ π‘₯ β†’ 𝒫 (cardβ€˜π‘₯) β‰ˆ 𝒫 π‘₯)
31 ennum 9891 . . . . 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 3139 . 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 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  βˆƒwrex 3070   βŠ† wss 3914  π’« cpw 4564   class class class wbr 5109  dom cdm 5637  Oncon0 6321  β€˜cfv 6500  Ο‰com 7806   β‰ˆ cen 8886  Fincfn 8889  cardccrd 9879  β„΅cale 9880
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 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-inf2 9585
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-se 5593  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-isom 6509  df-riota 7317  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-1st 7925  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-2o 8417  df-er 8654  df-map 8773  df-en 8890  df-dom 8891  df-sdom 8892  df-fin 8893  df-oi 9454  df-har 9501  df-card 9883  df-aleph 9884
This theorem is referenced by:  dfac12  10093
  Copyright terms: Public domain W3C validator