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Theorem dfac12k 10141
Description: Equivalence of dfac12 10143 and dfac12a 10142, 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 10063 . . . 4 (β„΅β€˜π‘¦) ∈ On
2 pweq 4616 . . . . . 6 (π‘₯ = (β„΅β€˜π‘¦) β†’ 𝒫 π‘₯ = 𝒫 (β„΅β€˜π‘¦))
32eleq1d 2818 . . . . 5 (π‘₯ = (β„΅β€˜π‘¦) β†’ (𝒫 π‘₯ ∈ dom card ↔ 𝒫 (β„΅β€˜π‘¦) ∈ dom card))
43rspcv 3608 . . . 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 3150 . 2 (βˆ€π‘₯ ∈ On 𝒫 π‘₯ ∈ dom card β†’ βˆ€π‘¦ ∈ On 𝒫 (β„΅β€˜π‘¦) ∈ dom card)
7 omelon 9640 . . . . . . 7 Ο‰ ∈ On
8 cardon 9938 . . . . . . 7 (cardβ€˜π‘₯) ∈ On
9 ontri1 6398 . . . . . . 7 ((Ο‰ ∈ On ∧ (cardβ€˜π‘₯) ∈ On) β†’ (Ο‰ βŠ† (cardβ€˜π‘₯) ↔ Β¬ (cardβ€˜π‘₯) ∈ Ο‰))
107, 8, 9mp2an 690 . . . . . 6 (Ο‰ βŠ† (cardβ€˜π‘₯) ↔ Β¬ (cardβ€˜π‘₯) ∈ Ο‰)
11 cardidm 9953 . . . . . . . 8 (cardβ€˜(cardβ€˜π‘₯)) = (cardβ€˜π‘₯)
12 cardalephex 10084 . . . . . . . 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 4616 . . . . . . . . . . . 12 ((cardβ€˜π‘₯) = (β„΅β€˜π‘¦) β†’ 𝒫 (cardβ€˜π‘₯) = 𝒫 (β„΅β€˜π‘¦))
1615eleq1d 2818 . . . . . . . . . . 11 ((cardβ€˜π‘₯) = (β„΅β€˜π‘¦) β†’ (𝒫 (cardβ€˜π‘₯) ∈ dom card ↔ 𝒫 (β„΅β€˜π‘¦) ∈ dom card))
1716biimparc 480 . . . . . . . . . 10 ((𝒫 (β„΅β€˜π‘¦) ∈ dom card ∧ (cardβ€˜π‘₯) = (β„΅β€˜π‘¦)) β†’ 𝒫 (cardβ€˜π‘₯) ∈ dom card)
1817rexlimivw 3151 . . . . . . . . 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 413 . . . . . . 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 9166 . . . . . . 7 ((cardβ€˜π‘₯) ∈ Ο‰ β†’ (cardβ€˜π‘₯) ∈ Fin)
24 pwfi 9177 . . . . . . 7 ((cardβ€˜π‘₯) ∈ Fin ↔ 𝒫 (cardβ€˜π‘₯) ∈ Fin)
2523, 24sylib 217 . . . . . 6 ((cardβ€˜π‘₯) ∈ Ο‰ β†’ 𝒫 (cardβ€˜π‘₯) ∈ Fin)
26 finnum 9942 . . . . . 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 9950 . . . . 5 (π‘₯ ∈ On β†’ (cardβ€˜π‘₯) β‰ˆ π‘₯)
30 pwen 9149 . . . . 5 ((cardβ€˜π‘₯) β‰ˆ π‘₯ β†’ 𝒫 (cardβ€˜π‘₯) β‰ˆ 𝒫 π‘₯)
31 ennum 9941 . . . . 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 3145 . 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 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  βˆƒwrex 3070   βŠ† wss 3948  π’« cpw 4602   class class class wbr 5148  dom cdm 5676  Oncon0 6364  β€˜cfv 6543  Ο‰com 7854   β‰ˆ cen 8935  Fincfn 8938  cardccrd 9929  β„΅cale 9930
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724  ax-inf2 9635
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-1o 8465  df-2o 8466  df-er 8702  df-map 8821  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-oi 9504  df-har 9551  df-card 9933  df-aleph 9934
This theorem is referenced by:  dfac12  10143
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