Theorem dfac12k 10108

Description: Equivalence of dfac12 10110 and dfac12a 10109, 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 10029	. . . 4 ⊢ (ℵ‘𝑦) ∈ On
2		pweq 4580	. . . . . 6 ⊢ (𝑥 = (ℵ‘𝑦) → 𝒫 𝑥 = 𝒫 (ℵ‘𝑦))
3	2	eleq1d 2814	. . . . 5 ⊢ (𝑥 = (ℵ‘𝑦) → (𝒫 𝑥 ∈ dom card ↔ 𝒫 (ℵ‘𝑦) ∈ dom card))
4	3	rspcv 3587	. . . 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 3130	. 2 ⊢ (∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card → ∀𝑦 ∈ On 𝒫 (ℵ‘𝑦) ∈ dom card)
7		omelon 9606	. . . . . . 7 ⊢ ω ∈ On
8		cardon 9904	. . . . . . 7 ⊢ (card‘𝑥) ∈ On
9		ontri1 6369	. . . . . . 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 9919	. . . . . . . 8 ⊢ (card‘(card‘𝑥)) = (card‘𝑥)
12		cardalephex 10050	. . . . . . . 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 3095	. . . . . . . . 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 4580	. . . . . . . . . . . 12 ⊢ ((card‘𝑥) = (ℵ‘𝑦) → 𝒫 (card‘𝑥) = 𝒫 (ℵ‘𝑦))
16	15	eleq1d 2814	. . . . . . . . . . 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 3131	. . . . . . . . 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 9137	. . . . . . 7 ⊢ ((card‘𝑥) ∈ ω → (card‘𝑥) ∈ Fin)
24		pwfi 9275	. . . . . . 7 ⊢ ((card‘𝑥) ∈ Fin ↔ 𝒫 (card‘𝑥) ∈ Fin)
25	23, 24	sylib 218	. . . . . 6 ⊢ ((card‘𝑥) ∈ ω → 𝒫 (card‘𝑥) ∈ Fin)
26		finnum 9908	. . . . . 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 9916	. . . . 5 ⊢ (𝑥 ∈ On → (card‘𝑥) ≈ 𝑥)
30		pwen 9120	. . . . 5 ⊢ ((card‘𝑥) ≈ 𝑥 → 𝒫 (card‘𝑥) ≈ 𝒫 𝑥)
31		ennum 9907	. . . . 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 3125	. 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 2109  ∀wral 3045  ∃wrex 3054   ⊆ wss 3917  𝒫 cpw 4566   class class class wbr 5110  dom cdm 5641  Oncon0 6335  ‘cfv 6514  ωcom 7845   ≈ cen 8918  Fincfn 8921  cardccrd 9895  ℵcale 9896

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-inf2 9601

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 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-se 5595  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-isom 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-2o 8438  df-er 8674  df-map 8804  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-oi 9470  df-har 9517  df-card 9899  df-aleph 9900

This theorem is referenced by:  dfac12  10110