Theorem dfac12k 10162

Description: Equivalence of dfac12 10164 and dfac12a 10163, 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

Step	Hyp	Ref	Expression
1		alephon 10083	. . . 4 ⊢ (ℵ‘𝑦) ∈ On
2		pweq 4589	. . . . . 6 ⊢ (𝑥 = (ℵ‘𝑦) → 𝒫 𝑥 = 𝒫 (ℵ‘𝑦))
3	2	eleq1d 2819	. . . . 5 ⊢ (𝑥 = (ℵ‘𝑦) → (𝒫 𝑥 ∈ dom card ↔ 𝒫 (ℵ‘𝑦) ∈ dom card))
4	3	rspcv 3597	. . . 4 ⊢ ((ℵ‘𝑦) ∈ On → (∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card → 𝒫 (ℵ‘𝑦) ∈ dom card))
5	1, 4	ax-mp 5	. . 3 ⊢ (∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card → 𝒫 (ℵ‘𝑦) ∈ dom card)
6	5	ralrimivw 3136	. 2 ⊢ (∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card → ∀𝑦 ∈ On 𝒫 (ℵ‘𝑦) ∈ dom card)
7		omelon 9660	. . . . . . 7 ⊢ ω ∈ On
8		cardon 9958	. . . . . . 7 ⊢ (card‘𝑥) ∈ On
9		ontri1 6386	. . . . . . 7 ⊢ ((ω ∈ On ∧ (card‘𝑥) ∈ On) → (ω ⊆ (card‘𝑥) ↔ ¬ (card‘𝑥) ∈ ω))
10	7, 8, 9	mp2an 692	. . . . . 6 ⊢ (ω ⊆ (card‘𝑥) ↔ ¬ (card‘𝑥) ∈ ω)
11		cardidm 9973	. . . . . . . 8 ⊢ (card‘(card‘𝑥)) = (card‘𝑥)
12		cardalephex 10104	. . . . . . . 8 ⊢ (ω ⊆ (card‘𝑥) → ((card‘(card‘𝑥)) = (card‘𝑥) ↔ ∃𝑦 ∈ On (card‘𝑥) = (ℵ‘𝑦)))
13	11, 12	mpbii 233	. . . . . . 7 ⊢ (ω ⊆ (card‘𝑥) → ∃𝑦 ∈ On (card‘𝑥) = (ℵ‘𝑦))
14		r19.29 3101	. . . . . . . . 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 4589	. . . . . . . . . . . 12 ⊢ ((card‘𝑥) = (ℵ‘𝑦) → 𝒫 (card‘𝑥) = 𝒫 (ℵ‘𝑦))
16	15	eleq1d 2819	. . . . . . . . . . 11 ⊢ ((card‘𝑥) = (ℵ‘𝑦) → (𝒫 (card‘𝑥) ∈ dom card ↔ 𝒫 (ℵ‘𝑦) ∈ dom card))
17	16	biimparc 479	. . . . . . . . . 10 ⊢ ((𝒫 (ℵ‘𝑦) ∈ dom card ∧ (card‘𝑥) = (ℵ‘𝑦)) → 𝒫 (card‘𝑥) ∈ dom card)
18	17	rexlimivw 3137	. . . . . . . . 9 ⊢ (∃𝑦 ∈ On (𝒫 (ℵ‘𝑦) ∈ dom card ∧ (card‘𝑥) = (ℵ‘𝑦)) → 𝒫 (card‘𝑥) ∈ dom card)
19	14, 18	syl 17	. . . . . . . 8 ⊢ ((∀𝑦 ∈ On 𝒫 (ℵ‘𝑦) ∈ dom card ∧ ∃𝑦 ∈ On (card‘𝑥) = (ℵ‘𝑦)) → 𝒫 (card‘𝑥) ∈ dom card)
20	19	ex 412	. . . . . . 7 ⊢ (∀𝑦 ∈ On 𝒫 (ℵ‘𝑦) ∈ dom card → (∃𝑦 ∈ On (card‘𝑥) = (ℵ‘𝑦) → 𝒫 (card‘𝑥) ∈ dom card))
21	13, 20	syl5 34	. . . . . 6 ⊢ (∀𝑦 ∈ On 𝒫 (ℵ‘𝑦) ∈ dom card → (ω ⊆ (card‘𝑥) → 𝒫 (card‘𝑥) ∈ dom card))
22	10, 21	biimtrrid 243	. . . . 5 ⊢ (∀𝑦 ∈ On 𝒫 (ℵ‘𝑦) ∈ dom card → (¬ (card‘𝑥) ∈ ω → 𝒫 (card‘𝑥) ∈ dom card))
23		nnfi 9181	. . . . . . 7 ⊢ ((card‘𝑥) ∈ ω → (card‘𝑥) ∈ Fin)
24		pwfi 9329	. . . . . . 7 ⊢ ((card‘𝑥) ∈ Fin ↔ 𝒫 (card‘𝑥) ∈ Fin)
25	23, 24	sylib 218	. . . . . 6 ⊢ ((card‘𝑥) ∈ ω → 𝒫 (card‘𝑥) ∈ Fin)
26		finnum 9962	. . . . . 6 ⊢ (𝒫 (card‘𝑥) ∈ Fin → 𝒫 (card‘𝑥) ∈ dom card)
27	25, 26	syl 17	. . . . 5 ⊢ ((card‘𝑥) ∈ ω → 𝒫 (card‘𝑥) ∈ dom card)
28	22, 27	pm2.61d2 181	. . . 4 ⊢ (∀𝑦 ∈ On 𝒫 (ℵ‘𝑦) ∈ dom card → 𝒫 (card‘𝑥) ∈ dom card)
29		oncardid 9970	. . . . 5 ⊢ (𝑥 ∈ On → (card‘𝑥) ≈ 𝑥)
30		pwen 9164	. . . . 5 ⊢ ((card‘𝑥) ≈ 𝑥 → 𝒫 (card‘𝑥) ≈ 𝒫 𝑥)
31		ennum 9961	. . . . 5 ⊢ (𝒫 (card‘𝑥) ≈ 𝒫 𝑥 → (𝒫 (card‘𝑥) ∈ dom card ↔ 𝒫 𝑥 ∈ dom card))
32	29, 30, 31	3syl 18	. . . 4 ⊢ (𝑥 ∈ On → (𝒫 (card‘𝑥) ∈ dom card ↔ 𝒫 𝑥 ∈ dom card))
33	28, 32	syl5ibcom 245	. . 3 ⊢ (∀𝑦 ∈ On 𝒫 (ℵ‘𝑦) ∈ dom card → (𝑥 ∈ On → 𝒫 𝑥 ∈ dom card))
34	33	ralrimiv 3131	. 2 ⊢ (∀𝑦 ∈ On 𝒫 (ℵ‘𝑦) ∈ dom card → ∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card)
35	6, 34	impbii 209	1 ⊢ (∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card ↔ ∀𝑦 ∈ On 𝒫 (ℵ‘𝑦) ∈ dom card)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3051  ∃wrex 3060   ⊆ wss 3926  𝒫 cpw 4575   class class class wbr 5119  dom cdm 5654  Oncon0 6352  ‘cfv 6531  ωcom 7861   ≈ cen 8956  Fincfn 8959  cardccrd 9949  ℵcale 9950

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 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-inf2 9655

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-se 5607  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-isom 6540  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7862  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-2o 8481  df-er 8719  df-map 8842  df-en 8960  df-dom 8961  df-sdom 8962  df-fin 8963  df-oi 9524  df-har 9571  df-card 9953  df-aleph 9954

This theorem is referenced by:  dfac12  10164