Theorem dfac12k 10186

Description: Equivalence of dfac12 10188 and dfac12a 10187, 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 10107	. . . 4 ⊢ (ℵ‘𝑦) ∈ On
2		pweq 4619	. . . . . 6 ⊢ (𝑥 = (ℵ‘𝑦) → 𝒫 𝑥 = 𝒫 (ℵ‘𝑦))
3	2	eleq1d 2824	. . . . 5 ⊢ (𝑥 = (ℵ‘𝑦) → (𝒫 𝑥 ∈ dom card ↔ 𝒫 (ℵ‘𝑦) ∈ dom card))
4	3	rspcv 3618	. . . 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 3148	. 2 ⊢ (∀𝑥 ∈ On 𝒫 𝑥 ∈ dom card → ∀𝑦 ∈ On 𝒫 (ℵ‘𝑦) ∈ dom card)
7		omelon 9684	. . . . . . 7 ⊢ ω ∈ On
8		cardon 9982	. . . . . . 7 ⊢ (card‘𝑥) ∈ On
9		ontri1 6420	. . . . . . 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 9997	. . . . . . . 8 ⊢ (card‘(card‘𝑥)) = (card‘𝑥)
12		cardalephex 10128	. . . . . . . 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 3112	. . . . . . . . 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 4619	. . . . . . . . . . . 12 ⊢ ((card‘𝑥) = (ℵ‘𝑦) → 𝒫 (card‘𝑥) = 𝒫 (ℵ‘𝑦))
16	15	eleq1d 2824	. . . . . . . . . . 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 3149	. . . . . . . . 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 9206	. . . . . . 7 ⊢ ((card‘𝑥) ∈ ω → (card‘𝑥) ∈ Fin)
24		pwfi 9355	. . . . . . 7 ⊢ ((card‘𝑥) ∈ Fin ↔ 𝒫 (card‘𝑥) ∈ Fin)
25	23, 24	sylib 218	. . . . . 6 ⊢ ((card‘𝑥) ∈ ω → 𝒫 (card‘𝑥) ∈ Fin)
26		finnum 9986	. . . . . 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 9994	. . . . 5 ⊢ (𝑥 ∈ On → (card‘𝑥) ≈ 𝑥)
30		pwen 9189	. . . . 5 ⊢ ((card‘𝑥) ≈ 𝑥 → 𝒫 (card‘𝑥) ≈ 𝒫 𝑥)
31		ennum 9985	. . . . 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 3143	. 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 1537   ∈ wcel 2106  ∀wral 3059  ∃wrex 3068   ⊆ wss 3963  𝒫 cpw 4605   class class class wbr 5148  dom cdm 5689  Oncon0 6386  ‘cfv 6563  ωcom 7887   ≈ cen 8981  Fincfn 8984  cardccrd 9973  ℵcale 9974

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-oi 9548  df-har 9595  df-card 9977  df-aleph 9978

This theorem is referenced by:  dfac12  10188