Theorem dfac12k 10153

Description: Equivalence of dfac12 10155 and dfac12a 10154, 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 10075	. . . 4 ⊢ (ℵ‘𝑦) ∈ On
2		pweq 4571	. . . . . 6 ⊢ (𝑥 = (ℵ‘𝑦) → 𝒫 𝑥 = 𝒫 (ℵ‘𝑦))
3	2	eleq1d 2845	. . . . 5 ⊢ (𝑥 = (ℵ‘𝑦) → (𝒫 𝑥 ∈ dom card ↔ 𝒫 (ℵ‘𝑦) ∈ dom card))
4	3	rspcv 3572	. . . 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 3158	. 2 ⊢ (∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card → ∀𝑦 ∈ On 𝒫 (ℵ‘𝑦) ∈ dom card)
7		omelon 9628	. . . . . . 7 ⊢ ω ∈ On
8		cardon 9952	. . . . . . 7 ⊢ (card‘𝑥) ∈ On
9		ontri1 6392	. . . . . . 7 ⊢ ((ω ∈ On ∧ (card‘𝑥) ∈ On) → (ω ⊆ (card‘𝑥) ↔ ¬ (card‘𝑥) ∈ ω))
10	7, 8, 9	mp2an 705	. . . . . 6 ⊢ (ω ⊆ (card‘𝑥) ↔ ¬ (card‘𝑥) ∈ ω)
11		cardidm 9967	. . . . . . . 8 ⊢ (card‘(card‘𝑥)) = (card‘𝑥)
12		cardalephex 10096	. . . . . . . 8 ⊢ (ω ⊆ (card‘𝑥) → ((card‘(card‘𝑥)) = (card‘𝑥) ↔ ∃𝑦 ∈ On (card‘𝑥) = (ℵ‘𝑦)))
13	11, 12	mpbii 236	. . . . . . 7 ⊢ (ω ⊆ (card‘𝑥) → ∃𝑦 ∈ On (card‘𝑥) = (ℵ‘𝑦))
14		r19.29 3125	. . . . . . . . 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 4571	. . . . . . . . . . . 12 ⊢ ((card‘𝑥) = (ℵ‘𝑦) → 𝒫 (card‘𝑥) = 𝒫 (ℵ‘𝑦))
16	15	eleq1d 2845	. . . . . . . . . . 11 ⊢ ((card‘𝑥) = (ℵ‘𝑦) → (𝒫 (card‘𝑥) ∈ dom card ↔ 𝒫 (ℵ‘𝑦) ∈ dom card))
17	16	biimparc 485	. . . . . . . . . 10 ⊢ ((𝒫 (ℵ‘𝑦) ∈ dom card ∧ (card‘𝑥) = (ℵ‘𝑦)) → 𝒫 (card‘𝑥) ∈ dom card)
18	17	rexlimivw 3159	. . . . . . . . 9 ⊢ (∃𝑦 ∈ On (𝒫 (ℵ‘𝑦) ∈ dom card ∧ (card‘𝑥) = (ℵ‘𝑦)) → 𝒫 (card‘𝑥) ∈ dom card)
19	14, 18	syl 18	. . . . . . . 8 ⊢ ((∀𝑦 ∈ On 𝒫 (ℵ‘𝑦) ∈ dom card ∧ ∃𝑦 ∈ On (card‘𝑥) = (ℵ‘𝑦)) → 𝒫 (card‘𝑥) ∈ dom card)
20	19	ex 418	. . . . . . 7 ⊢ (∀𝑦 ∈ On 𝒫 (ℵ‘𝑦) ∈ dom card → (∃𝑦 ∈ On (card‘𝑥) = (ℵ‘𝑦) → 𝒫 (card‘𝑥) ∈ dom card))
21	13, 20	syl5 35	. . . . . 6 ⊢ (∀𝑦 ∈ On 𝒫 (ℵ‘𝑦) ∈ dom card → (ω ⊆ (card‘𝑥) → 𝒫 (card‘𝑥) ∈ dom card))
22	10, 21	biimtrrid 246	. . . . 5 ⊢ (∀𝑦 ∈ On 𝒫 (ℵ‘𝑦) ∈ dom card → (¬ (card‘𝑥) ∈ ω → 𝒫 (card‘𝑥) ∈ dom card))
23		nnfi 9165	. . . . . . 7 ⊢ ((card‘𝑥) ∈ ω → (card‘𝑥) ∈ Fin)
24		pwfi 9291	. . . . . . 7 ⊢ ((card‘𝑥) ∈ Fin ↔ 𝒫 (card‘𝑥) ∈ Fin)
25	23, 24	sylib 221	. . . . . 6 ⊢ ((card‘𝑥) ∈ ω → 𝒫 (card‘𝑥) ∈ Fin)
26		finnum 9956	. . . . . 6 ⊢ (𝒫 (card‘𝑥) ∈ Fin → 𝒫 (card‘𝑥) ∈ dom card)
27	25, 26	syl 18	. . . . 5 ⊢ ((card‘𝑥) ∈ ω → 𝒫 (card‘𝑥) ∈ dom card)
28	22, 27	pm2.61d2 183	. . . 4 ⊢ (∀𝑦 ∈ On 𝒫 (ℵ‘𝑦) ∈ dom card → 𝒫 (card‘𝑥) ∈ dom card)
29		oncardid 9964	. . . . 5 ⊢ (𝑥 ∈ On → (card‘𝑥) ≈ 𝑥)
30		pwen 9151	. . . . 5 ⊢ ((card‘𝑥) ≈ 𝑥 → 𝒫 (card‘𝑥) ≈ 𝒫 𝑥)
31		ennum 9955	. . . . 5 ⊢ (𝒫 (card‘𝑥) ≈ 𝒫 𝑥 → (𝒫 (card‘𝑥) ∈ dom card ↔ 𝒫 𝑥 ∈ dom card))
32	29, 30, 31	3syl 19	. . . 4 ⊢ (𝑥 ∈ On → (𝒫 (card‘𝑥) ∈ dom card ↔ 𝒫 𝑥 ∈ dom card))
33	28, 32	syl5ibcom 248	. . 3 ⊢ (∀𝑦 ∈ On 𝒫 (ℵ‘𝑦) ∈ dom card → (𝑥 ∈ On → 𝒫 𝑥 ∈ dom card))
34	33	ralrimiv 3153	. 2 ⊢ (∀𝑦 ∈ On 𝒫 (ℵ‘𝑦) ∈ dom card → ∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card)
35	6, 34	impbii 212	1 ⊢ (∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card ↔ ∀𝑦 ∈ On 𝒫 (ℵ‘𝑦) ∈ dom card)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076  ∃wrex 3086   ⊆ wss 3899  𝒫 cpw 4557   class class class wbr 5103  dom cdm 5655  Oncon0 6357  ‘cfv 6533  ωcom 7863   ≈ cen 8952  Fincfn 8955  cardccrd 9943  ℵcale 9944

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737  ax-inf2 9623

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8281  df-wrecs 8312  df-recs 8361  df-rdg 8400  df-1o 8458  df-2o 8459  df-er 8699  df-map 8831  df-en 8956  df-dom 8957  df-sdom 8958  df-fin 8959  df-oi 9485  df-har 9532  df-card 9947  df-aleph 9948

This theorem is used by:  dfac12  10155