Theorem dfac12k 10039

Description: Equivalence of dfac12 10041 and dfac12a 10040, 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 9960	. . . 4 ⊢ (ℵ‘𝑦) ∈ On
2		pweq 4561	. . . . . 6 ⊢ (𝑥 = (ℵ‘𝑦) → 𝒫 𝑥 = 𝒫 (ℵ‘𝑦))
3	2	eleq1d 2816	. . . . 5 ⊢ (𝑥 = (ℵ‘𝑦) → (𝒫 𝑥 ∈ dom card ↔ 𝒫 (ℵ‘𝑦) ∈ dom card))
4	3	rspcv 3568	. . . 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 3128	. 2 ⊢ (∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card → ∀𝑦 ∈ On 𝒫 (ℵ‘𝑦) ∈ dom card)
7		omelon 9536	. . . . . . 7 ⊢ ω ∈ On
8		cardon 9837	. . . . . . 7 ⊢ (card‘𝑥) ∈ On
9		ontri1 6340	. . . . . . 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 9852	. . . . . . . 8 ⊢ (card‘(card‘𝑥)) = (card‘𝑥)
12		cardalephex 9981	. . . . . . . 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 4561	. . . . . . . . . . . 12 ⊢ ((card‘𝑥) = (ℵ‘𝑦) → 𝒫 (card‘𝑥) = 𝒫 (ℵ‘𝑦))
16	15	eleq1d 2816	. . . . . . . . . . 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 3129	. . . . . . . . 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 9077	. . . . . . 7 ⊢ ((card‘𝑥) ∈ ω → (card‘𝑥) ∈ Fin)
24		pwfi 9203	. . . . . . 7 ⊢ ((card‘𝑥) ∈ Fin ↔ 𝒫 (card‘𝑥) ∈ Fin)
25	23, 24	sylib 218	. . . . . 6 ⊢ ((card‘𝑥) ∈ ω → 𝒫 (card‘𝑥) ∈ Fin)
26		finnum 9841	. . . . . 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 9849	. . . . 5 ⊢ (𝑥 ∈ On → (card‘𝑥) ≈ 𝑥)
30		pwen 9063	. . . . 5 ⊢ ((card‘𝑥) ≈ 𝑥 → 𝒫 (card‘𝑥) ≈ 𝒫 𝑥)
31		ennum 9840	. . . . 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 3123	. 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 1541   ∈ wcel 2111  ∀wral 3047  ∃wrex 3056   ⊆ wss 3897  𝒫 cpw 4547   class class class wbr 5089  dom cdm 5614  Oncon0 6306  ‘cfv 6481  ωcom 7796   ≈ cen 8866  Fincfn 8869  cardccrd 9828  ℵcale 9829

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-inf2 9531

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-int 4896  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-se 5568  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-isom 6490  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-1o 8385  df-2o 8386  df-er 8622  df-map 8752  df-en 8870  df-dom 8871  df-sdom 8872  df-fin 8873  df-oi 9396  df-har 9443  df-card 9832  df-aleph 9833

This theorem is referenced by:  dfac12  10041