Theorem nenpw1pwlem2 6086

Description: Lemma for nenpw1pw 6087. Establish the main theorem with an extra hypothesis. (Contributed by SF, 10-Mar-2015.)

Hypothesis

Ref	Expression
nenpw1pwlem2.1	⊢ S = {x ∈ A ∣ ¬ x ∈ (r ‘{x})}

Assertion

Ref	Expression
nenpw1pwlem2	⊢ ¬ 𝒫1A ≈ 𝒫 A

Distinct variable group:   A,r,x

Allowed substitution hints:   S(x, r)

Proof of Theorem nenpw1pwlem2

Dummy variables u y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pm5.19 349	. . . . 5 ⊢ ¬ (y ∈ S ↔ ¬ y ∈ S)
2	1	a1i 10	. . . 4 ⊢ (y ∈ A → ¬ (y ∈ S ↔ ¬ y ∈ S))
3	2	nrex 2717	. . 3 ⊢ ¬ ∃y ∈ A (y ∈ S ↔ ¬ y ∈ S)
4	3	nex 1555	. 2 ⊢ ¬ ∃r∃y ∈ A (y ∈ S ↔ ¬ y ∈ S)
5		bren 6031	. . 3 ⊢ (𝒫1A ≈ 𝒫 A ↔ ∃r r:𝒫1A–1-1-onto→𝒫 A)
6		f1odm 5291	. . . . . . . . 9 ⊢ (r:𝒫1A–1-1-onto→𝒫 A → dom r = 𝒫1A)
7		vex 2863	. . . . . . . . . 10 ⊢ r ∈ V
8	7	dmex 5107	. . . . . . . . 9 ⊢ dom r ∈ V
9	6, 8	syl6eqelr 2442	. . . . . . . 8 ⊢ (r:𝒫1A–1-1-onto→𝒫 A → 𝒫1A ∈ V)
10		pw1exb 4327	. . . . . . . 8 ⊢ (𝒫1A ∈ V ↔ A ∈ V)
11	9, 10	sylib 188	. . . . . . 7 ⊢ (r:𝒫1A–1-1-onto→𝒫 A → A ∈ V)
12		nenpw1pwlem2.1	. . . . . . . 8 ⊢ S = {x ∈ A ∣ ¬ x ∈ (r ‘{x})}
13	12	nenpw1pwlem1 6085	. . . . . . 7 ⊢ (A ∈ V → S ∈ V)
14	11, 13	syl 15	. . . . . 6 ⊢ (r:𝒫1A–1-1-onto→𝒫 A → S ∈ V)
15		ssrab2 3352	. . . . . . . 8 ⊢ {x ∈ A ∣ ¬ x ∈ (r ‘{x})} ⊆ A
16	12, 15	eqsstri 3302	. . . . . . 7 ⊢ S ⊆ A
17		elpwg 3730	. . . . . . 7 ⊢ (S ∈ V → (S ∈ 𝒫 A ↔ S ⊆ A))
18	16, 17	mpbiri 224	. . . . . 6 ⊢ (S ∈ V → S ∈ 𝒫 A)
19	14, 18	syl 15	. . . . 5 ⊢ (r:𝒫1A–1-1-onto→𝒫 A → S ∈ 𝒫 A)
20		f1ofo 5294	. . . . . . . 8 ⊢ (r:𝒫1A–1-1-onto→𝒫 A → r:𝒫1A–onto→𝒫 A)
21		forn 5273	. . . . . . . 8 ⊢ (r:𝒫1A–onto→𝒫 A → ran r = 𝒫 A)
22	20, 21	syl 15	. . . . . . 7 ⊢ (r:𝒫1A–1-1-onto→𝒫 A → ran r = 𝒫 A)
23	22	eleq2d 2420	. . . . . 6 ⊢ (r:𝒫1A–1-1-onto→𝒫 A → (S ∈ ran r ↔ S ∈ 𝒫 A))
24		elrn 4897	. . . . . . 7 ⊢ (S ∈ ran r ↔ ∃u urS)
25		breldm 4912	. . . . . . . . . . . 12 ⊢ (urS → u ∈ dom r)
26	25	adantl 452	. . . . . . . . . . 11 ⊢ ((r:𝒫1A–1-1-onto→𝒫 A ∧ urS) → u ∈ dom r)
27	6	adantr 451	. . . . . . . . . . 11 ⊢ ((r:𝒫1A–1-1-onto→𝒫 A ∧ urS) → dom r = 𝒫1A)
28	26, 27	eleqtrd 2429	. . . . . . . . . 10 ⊢ ((r:𝒫1A–1-1-onto→𝒫 A ∧ urS) → u ∈ 𝒫1A)
29		elpw1 4145	. . . . . . . . . . 11 ⊢ (u ∈ 𝒫1A ↔ ∃y ∈ A u = {y})
30		breq1 4643	. . . . . . . . . . . . . . . 16 ⊢ (u = {y} → (urS ↔ {y}rS))
31	30	3anbi2d 1257	. . . . . . . . . . . . . . 15 ⊢ (u = {y} → ((r:𝒫1A–1-1-onto→𝒫 A ∧ urS ∧ y ∈ A) ↔ (r:𝒫1A–1-1-onto→𝒫 A ∧ {y}rS ∧ y ∈ A)))
32		id 19	. . . . . . . . . . . . . . . . . . 19 ⊢ (x = y → x = y)
33		sneq 3745	. . . . . . . . . . . . . . . . . . . 20 ⊢ (x = y → {x} = {y})
34	33	fveq2d 5333	. . . . . . . . . . . . . . . . . . 19 ⊢ (x = y → (r ‘{x}) = (r ‘{y}))
35	32, 34	eleq12d 2421	. . . . . . . . . . . . . . . . . 18 ⊢ (x = y → (x ∈ (r ‘{x}) ↔ y ∈ (r ‘{y})))
36	35	notbid 285	. . . . . . . . . . . . . . . . 17 ⊢ (x = y → (¬ x ∈ (r ‘{x}) ↔ ¬ y ∈ (r ‘{y})))
37	36, 12	elrab2 2997	. . . . . . . . . . . . . . . 16 ⊢ (y ∈ S ↔ (y ∈ A ∧ ¬ y ∈ (r ‘{y})))
38		simp3 957	. . . . . . . . . . . . . . . . . 18 ⊢ ((r:𝒫1A–1-1-onto→𝒫 A ∧ {y}rS ∧ y ∈ A) → y ∈ A)
39	38	biantrurd 494	. . . . . . . . . . . . . . . . 17 ⊢ ((r:𝒫1A–1-1-onto→𝒫 A ∧ {y}rS ∧ y ∈ A) → (¬ y ∈ (r ‘{y}) ↔ (y ∈ A ∧ ¬ y ∈ (r ‘{y}))))
40		simp2 956	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((r:𝒫1A–1-1-onto→𝒫 A ∧ {y}rS ∧ y ∈ A) → {y}rS)
41		f1ofn 5289	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (r:𝒫1A–1-1-onto→𝒫 A → r Fn 𝒫1A)
42	41	3ad2ant1 976	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((r:𝒫1A–1-1-onto→𝒫 A ∧ {y}rS ∧ y ∈ A) → r Fn 𝒫1A)
43		snelpw1 4147	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ({y} ∈ 𝒫1A ↔ y ∈ A)
44	43	biimpri 197	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (y ∈ A → {y} ∈ 𝒫1A)
45	44	3ad2ant3 978	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((r:𝒫1A–1-1-onto→𝒫 A ∧ {y}rS ∧ y ∈ A) → {y} ∈ 𝒫1A)
46		fnbrfvb 5359	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((r Fn 𝒫1A ∧ {y} ∈ 𝒫1A) → ((r ‘{y}) = S ↔ {y}rS))
47	42, 45, 46	syl2anc 642	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((r:𝒫1A–1-1-onto→𝒫 A ∧ {y}rS ∧ y ∈ A) → ((r ‘{y}) = S ↔ {y}rS))
48	40, 47	mpbird 223	. . . . . . . . . . . . . . . . . . 19 ⊢ ((r:𝒫1A–1-1-onto→𝒫 A ∧ {y}rS ∧ y ∈ A) → (r ‘{y}) = S)
49	48	eleq2d 2420	. . . . . . . . . . . . . . . . . 18 ⊢ ((r:𝒫1A–1-1-onto→𝒫 A ∧ {y}rS ∧ y ∈ A) → (y ∈ (r ‘{y}) ↔ y ∈ S))
50	49	notbid 285	. . . . . . . . . . . . . . . . 17 ⊢ ((r:𝒫1A–1-1-onto→𝒫 A ∧ {y}rS ∧ y ∈ A) → (¬ y ∈ (r ‘{y}) ↔ ¬ y ∈ S))
51	39, 50	bitr3d 246	. . . . . . . . . . . . . . . 16 ⊢ ((r:𝒫1A–1-1-onto→𝒫 A ∧ {y}rS ∧ y ∈ A) → ((y ∈ A ∧ ¬ y ∈ (r ‘{y})) ↔ ¬ y ∈ S))
52	37, 51	syl5bb 248	. . . . . . . . . . . . . . 15 ⊢ ((r:𝒫1A–1-1-onto→𝒫 A ∧ {y}rS ∧ y ∈ A) → (y ∈ S ↔ ¬ y ∈ S))
53	31, 52	syl6bi 219	. . . . . . . . . . . . . 14 ⊢ (u = {y} → ((r:𝒫1A–1-1-onto→𝒫 A ∧ urS ∧ y ∈ A) → (y ∈ S ↔ ¬ y ∈ S)))
54	53	com12 27	. . . . . . . . . . . . 13 ⊢ ((r:𝒫1A–1-1-onto→𝒫 A ∧ urS ∧ y ∈ A) → (u = {y} → (y ∈ S ↔ ¬ y ∈ S)))
55	54	3expa 1151	. . . . . . . . . . . 12 ⊢ (((r:𝒫1A–1-1-onto→𝒫 A ∧ urS) ∧ y ∈ A) → (u = {y} → (y ∈ S ↔ ¬ y ∈ S)))
56	55	reximdva 2727	. . . . . . . . . . 11 ⊢ ((r:𝒫1A–1-1-onto→𝒫 A ∧ urS) → (∃y ∈ A u = {y} → ∃y ∈ A (y ∈ S ↔ ¬ y ∈ S)))
57	29, 56	syl5bi 208	. . . . . . . . . 10 ⊢ ((r:𝒫1A–1-1-onto→𝒫 A ∧ urS) → (u ∈ 𝒫1A → ∃y ∈ A (y ∈ S ↔ ¬ y ∈ S)))
58	28, 57	mpd 14	. . . . . . . . 9 ⊢ ((r:𝒫1A–1-1-onto→𝒫 A ∧ urS) → ∃y ∈ A (y ∈ S ↔ ¬ y ∈ S))
59	58	ex 423	. . . . . . . 8 ⊢ (r:𝒫1A–1-1-onto→𝒫 A → (urS → ∃y ∈ A (y ∈ S ↔ ¬ y ∈ S)))
60	59	exlimdv 1636	. . . . . . 7 ⊢ (r:𝒫1A–1-1-onto→𝒫 A → (∃u urS → ∃y ∈ A (y ∈ S ↔ ¬ y ∈ S)))
61	24, 60	syl5bi 208	. . . . . 6 ⊢ (r:𝒫1A–1-1-onto→𝒫 A → (S ∈ ran r → ∃y ∈ A (y ∈ S ↔ ¬ y ∈ S)))
62	23, 61	sylbird 226	. . . . 5 ⊢ (r:𝒫1A–1-1-onto→𝒫 A → (S ∈ 𝒫 A → ∃y ∈ A (y ∈ S ↔ ¬ y ∈ S)))
63	19, 62	mpd 14	. . . 4 ⊢ (r:𝒫1A–1-1-onto→𝒫 A → ∃y ∈ A (y ∈ S ↔ ¬ y ∈ S))
64	63	eximi 1576	. . 3 ⊢ (∃r r:𝒫1A–1-1-onto→𝒫 A → ∃r∃y ∈ A (y ∈ S ↔ ¬ y ∈ S))
65	5, 64	sylbi 187	. 2 ⊢ (𝒫1A ≈ 𝒫 A → ∃r∃y ∈ A (y ∈ S ↔ ¬ y ∈ S))
66	4, 65	mto 167	1 ⊢ ¬ 𝒫1A ≈ 𝒫 A

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358   ∧ w3a 934  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2616  {crab 2619  Vcvv 2860   ⊆ wss 3258  𝒫 cpw 3723  {csn 3738  𝒫₁cpw1 4136   class class class wbr 4640  dom cdm 4773  ran crn 4774   Fn wfn 4777  –onto→wfo 4780  –1-1-onto→wf1o 4781   ‘cfv 4782   ≈ cen 6029

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-1st 4724  df-swap 4725  df-sset 4726  df-co 4727  df-ima 4728  df-id 4768  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788  df-res 4789  df-fun 4790  df-fn 4791  df-f 4792  df-f1 4793  df-fo 4794  df-f1o 4795  df-fv 4796  df-2nd 4798  df-fullfun 5769  df-en 6030

This theorem is used by:  nenpw1pw  6087