Theorem ustn0 24164

Description: The empty set is not an uniform structure. (Contributed by Thierry Arnoux, 3-Dec-2017.)

Assertion

Ref	Expression
ustn0	⊢ ¬ ∅ ∈ ∪ ran UnifOn

Proof of Theorem ustn0

Dummy variables 𝑣 𝑢 𝑤 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		noel 4318	. . . . 5 ⊢ ¬ (𝑥 × 𝑥) ∈ ∅
2		0ex 5282	. . . . . 6 ⊢ ∅ ∈ V
3		eleq2 2824	. . . . . 6 ⊢ (𝑢 = ∅ → ((𝑥 × 𝑥) ∈ 𝑢 ↔ (𝑥 × 𝑥) ∈ ∅))
4	2, 3	elab 3663	. . . . 5 ⊢ (∅ ∈ {𝑢 ∣ (𝑥 × 𝑥) ∈ 𝑢} ↔ (𝑥 × 𝑥) ∈ ∅)
5	1, 4	mtbir 323	. . . 4 ⊢ ¬ ∅ ∈ {𝑢 ∣ (𝑥 × 𝑥) ∈ 𝑢}
6		vex 3468	. . . . . . 7 ⊢ 𝑥 ∈ V
7		velpw 4585	. . . . . . . . . 10 ⊢ (𝑢 ∈ 𝒫 𝒫 (𝑥 × 𝑥) ↔ 𝑢 ⊆ 𝒫 (𝑥 × 𝑥))
8	7	abbii 2803	. . . . . . . . 9 ⊢ {𝑢 ∣ 𝑢 ∈ 𝒫 𝒫 (𝑥 × 𝑥)} = {𝑢 ∣ 𝑢 ⊆ 𝒫 (𝑥 × 𝑥)}
9		abid2 2873	. . . . . . . . . 10 ⊢ {𝑢 ∣ 𝑢 ∈ 𝒫 𝒫 (𝑥 × 𝑥)} = 𝒫 𝒫 (𝑥 × 𝑥)
10	6, 6	xpex 7752	. . . . . . . . . . . 12 ⊢ (𝑥 × 𝑥) ∈ V
11	10	pwex 5355	. . . . . . . . . . 11 ⊢ 𝒫 (𝑥 × 𝑥) ∈ V
12	11	pwex 5355	. . . . . . . . . 10 ⊢ 𝒫 𝒫 (𝑥 × 𝑥) ∈ V
13	9, 12	eqeltri 2831	. . . . . . . . 9 ⊢ {𝑢 ∣ 𝑢 ∈ 𝒫 𝒫 (𝑥 × 𝑥)} ∈ V
14	8, 13	eqeltrri 2832	. . . . . . . 8 ⊢ {𝑢 ∣ 𝑢 ⊆ 𝒫 (𝑥 × 𝑥)} ∈ V
15		simp1 1136	. . . . . . . . 9 ⊢ ((𝑢 ⊆ 𝒫 (𝑥 × 𝑥) ∧ (𝑥 × 𝑥) ∈ 𝑢 ∧ ∀𝑣 ∈ 𝑢 (∀𝑤 ∈ 𝒫 (𝑥 × 𝑥)(𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑢) ∧ ∀𝑤 ∈ 𝑢 (𝑣 ∩ 𝑤) ∈ 𝑢 ∧ (( I ↾ 𝑥) ⊆ 𝑣 ∧ ◡𝑣 ∈ 𝑢 ∧ ∃𝑤 ∈ 𝑢 (𝑤 ∘ 𝑤) ⊆ 𝑣))) → 𝑢 ⊆ 𝒫 (𝑥 × 𝑥))
16	15	ss2abi 4047	. . . . . . . 8 ⊢ {𝑢 ∣ (𝑢 ⊆ 𝒫 (𝑥 × 𝑥) ∧ (𝑥 × 𝑥) ∈ 𝑢 ∧ ∀𝑣 ∈ 𝑢 (∀𝑤 ∈ 𝒫 (𝑥 × 𝑥)(𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑢) ∧ ∀𝑤 ∈ 𝑢 (𝑣 ∩ 𝑤) ∈ 𝑢 ∧ (( I ↾ 𝑥) ⊆ 𝑣 ∧ ◡𝑣 ∈ 𝑢 ∧ ∃𝑤 ∈ 𝑢 (𝑤 ∘ 𝑤) ⊆ 𝑣)))} ⊆ {𝑢 ∣ 𝑢 ⊆ 𝒫 (𝑥 × 𝑥)}
17	14, 16	ssexi 5297	. . . . . . 7 ⊢ {𝑢 ∣ (𝑢 ⊆ 𝒫 (𝑥 × 𝑥) ∧ (𝑥 × 𝑥) ∈ 𝑢 ∧ ∀𝑣 ∈ 𝑢 (∀𝑤 ∈ 𝒫 (𝑥 × 𝑥)(𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑢) ∧ ∀𝑤 ∈ 𝑢 (𝑣 ∩ 𝑤) ∈ 𝑢 ∧ (( I ↾ 𝑥) ⊆ 𝑣 ∧ ◡𝑣 ∈ 𝑢 ∧ ∃𝑤 ∈ 𝑢 (𝑤 ∘ 𝑤) ⊆ 𝑣)))} ∈ V
18		df-ust 24144	. . . . . . . 8 ⊢ UnifOn = (𝑥 ∈ V ↦ {𝑢 ∣ (𝑢 ⊆ 𝒫 (𝑥 × 𝑥) ∧ (𝑥 × 𝑥) ∈ 𝑢 ∧ ∀𝑣 ∈ 𝑢 (∀𝑤 ∈ 𝒫 (𝑥 × 𝑥)(𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑢) ∧ ∀𝑤 ∈ 𝑢 (𝑣 ∩ 𝑤) ∈ 𝑢 ∧ (( I ↾ 𝑥) ⊆ 𝑣 ∧ ◡𝑣 ∈ 𝑢 ∧ ∃𝑤 ∈ 𝑢 (𝑤 ∘ 𝑤) ⊆ 𝑣)))})
19	18	fvmpt2 7002	. . . . . . 7 ⊢ ((𝑥 ∈ V ∧ {𝑢 ∣ (𝑢 ⊆ 𝒫 (𝑥 × 𝑥) ∧ (𝑥 × 𝑥) ∈ 𝑢 ∧ ∀𝑣 ∈ 𝑢 (∀𝑤 ∈ 𝒫 (𝑥 × 𝑥)(𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑢) ∧ ∀𝑤 ∈ 𝑢 (𝑣 ∩ 𝑤) ∈ 𝑢 ∧ (( I ↾ 𝑥) ⊆ 𝑣 ∧ ◡𝑣 ∈ 𝑢 ∧ ∃𝑤 ∈ 𝑢 (𝑤 ∘ 𝑤) ⊆ 𝑣)))} ∈ V) → (UnifOn‘𝑥) = {𝑢 ∣ (𝑢 ⊆ 𝒫 (𝑥 × 𝑥) ∧ (𝑥 × 𝑥) ∈ 𝑢 ∧ ∀𝑣 ∈ 𝑢 (∀𝑤 ∈ 𝒫 (𝑥 × 𝑥)(𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑢) ∧ ∀𝑤 ∈ 𝑢 (𝑣 ∩ 𝑤) ∈ 𝑢 ∧ (( I ↾ 𝑥) ⊆ 𝑣 ∧ ◡𝑣 ∈ 𝑢 ∧ ∃𝑤 ∈ 𝑢 (𝑤 ∘ 𝑤) ⊆ 𝑣)))})
20	6, 17, 19	mp2an 692	. . . . . 6 ⊢ (UnifOn‘𝑥) = {𝑢 ∣ (𝑢 ⊆ 𝒫 (𝑥 × 𝑥) ∧ (𝑥 × 𝑥) ∈ 𝑢 ∧ ∀𝑣 ∈ 𝑢 (∀𝑤 ∈ 𝒫 (𝑥 × 𝑥)(𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑢) ∧ ∀𝑤 ∈ 𝑢 (𝑣 ∩ 𝑤) ∈ 𝑢 ∧ (( I ↾ 𝑥) ⊆ 𝑣 ∧ ◡𝑣 ∈ 𝑢 ∧ ∃𝑤 ∈ 𝑢 (𝑤 ∘ 𝑤) ⊆ 𝑣)))}
21		simp2 1137	. . . . . . 7 ⊢ ((𝑢 ⊆ 𝒫 (𝑥 × 𝑥) ∧ (𝑥 × 𝑥) ∈ 𝑢 ∧ ∀𝑣 ∈ 𝑢 (∀𝑤 ∈ 𝒫 (𝑥 × 𝑥)(𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑢) ∧ ∀𝑤 ∈ 𝑢 (𝑣 ∩ 𝑤) ∈ 𝑢 ∧ (( I ↾ 𝑥) ⊆ 𝑣 ∧ ◡𝑣 ∈ 𝑢 ∧ ∃𝑤 ∈ 𝑢 (𝑤 ∘ 𝑤) ⊆ 𝑣))) → (𝑥 × 𝑥) ∈ 𝑢)
22	21	ss2abi 4047	. . . . . 6 ⊢ {𝑢 ∣ (𝑢 ⊆ 𝒫 (𝑥 × 𝑥) ∧ (𝑥 × 𝑥) ∈ 𝑢 ∧ ∀𝑣 ∈ 𝑢 (∀𝑤 ∈ 𝒫 (𝑥 × 𝑥)(𝑣 ⊆ 𝑤 → 𝑤 ∈ 𝑢) ∧ ∀𝑤 ∈ 𝑢 (𝑣 ∩ 𝑤) ∈ 𝑢 ∧ (( I ↾ 𝑥) ⊆ 𝑣 ∧ ◡𝑣 ∈ 𝑢 ∧ ∃𝑤 ∈ 𝑢 (𝑤 ∘ 𝑤) ⊆ 𝑣)))} ⊆ {𝑢 ∣ (𝑥 × 𝑥) ∈ 𝑢}
23	20, 22	eqsstri 4010	. . . . 5 ⊢ (UnifOn‘𝑥) ⊆ {𝑢 ∣ (𝑥 × 𝑥) ∈ 𝑢}
24	23	sseli 3959	. . . 4 ⊢ (∅ ∈ (UnifOn‘𝑥) → ∅ ∈ {𝑢 ∣ (𝑥 × 𝑥) ∈ 𝑢})
25	5, 24	mto 197	. . 3 ⊢ ¬ ∅ ∈ (UnifOn‘𝑥)
26	25	nex 1800	. 2 ⊢ ¬ ∃𝑥∅ ∈ (UnifOn‘𝑥)
27	18	funmpt2 6580	. . . 4 ⊢ Fun UnifOn
28		elunirn 7248	. . . 4 ⊢ (Fun UnifOn → (∅ ∈ ∪ ran UnifOn ↔ ∃𝑥 ∈ dom UnifOn∅ ∈ (UnifOn‘𝑥)))
29	27, 28	ax-mp 5	. . 3 ⊢ (∅ ∈ ∪ ran UnifOn ↔ ∃𝑥 ∈ dom UnifOn∅ ∈ (UnifOn‘𝑥))
30		ustfn 24145	. . . . 5 ⊢ UnifOn Fn V
31		fndm 6646	. . . . 5 ⊢ (UnifOn Fn V → dom UnifOn = V)
32	30, 31	ax-mp 5	. . . 4 ⊢ dom UnifOn = V
33	32	rexeqi 3308	. . 3 ⊢ (∃𝑥 ∈ dom UnifOn∅ ∈ (UnifOn‘𝑥) ↔ ∃𝑥 ∈ V ∅ ∈ (UnifOn‘𝑥))
34		rexv 3493	. . 3 ⊢ (∃𝑥 ∈ V ∅ ∈ (UnifOn‘𝑥) ↔ ∃𝑥∅ ∈ (UnifOn‘𝑥))
35	29, 33, 34	3bitri 297	. 2 ⊢ (∅ ∈ ∪ ran UnifOn ↔ ∃𝑥∅ ∈ (UnifOn‘𝑥))
36	26, 35	mtbir 323	1 ⊢ ¬ ∅ ∈ ∪ ran UnifOn

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {cab 2714  ∀wral 3052  ∃wrex 3061  Vcvv 3464   ∩ cin 3930   ⊆ wss 3931  ∅c0 4313  𝒫 cpw 4580  ∪ cuni 4888   I cid 5552   × cxp 5657  ^◡ccnv 5658  dom cdm 5659  ran crn 5660   ↾ cres 5661   ∘ ccom 5663  Fun wfun 6530   Fn wfn 6531  ‘cfv 6536  UnifOncust 24143

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 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-fv 6544  df-ust 24144

This theorem is referenced by: (None)