Theorem ustn0 24137

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 4287	. . . . 5 ⊢ ¬ (𝑥 × 𝑥) ∈ ∅
2		0ex 5247	. . . . . 6 ⊢ ∅ ∈ V
3		eleq2 2822	. . . . . 6 ⊢ (𝑢 = ∅ → ((𝑥 × 𝑥) ∈ 𝑢 ↔ (𝑥 × 𝑥) ∈ ∅))
4	2, 3	elab 3631	. . . . 5 ⊢ (∅ ∈ {𝑢 ∣ (𝑥 × 𝑥) ∈ 𝑢} ↔ (𝑥 × 𝑥) ∈ ∅)
5	1, 4	mtbir 323	. . . 4 ⊢ ¬ ∅ ∈ {𝑢 ∣ (𝑥 × 𝑥) ∈ 𝑢}
6		vex 3441	. . . . . . 7 ⊢ 𝑥 ∈ V
7		velpw 4554	. . . . . . . . . 10 ⊢ (𝑢 ∈ 𝒫 𝒫 (𝑥 × 𝑥) ↔ 𝑢 ⊆ 𝒫 (𝑥 × 𝑥))
8	7	abbii 2800	. . . . . . . . 9 ⊢ {𝑢 ∣ 𝑢 ∈ 𝒫 𝒫 (𝑥 × 𝑥)} = {𝑢 ∣ 𝑢 ⊆ 𝒫 (𝑥 × 𝑥)}
9		abid2 2870	. . . . . . . . . 10 ⊢ {𝑢 ∣ 𝑢 ∈ 𝒫 𝒫 (𝑥 × 𝑥)} = 𝒫 𝒫 (𝑥 × 𝑥)
10	6, 6	xpex 7692	. . . . . . . . . . . 12 ⊢ (𝑥 × 𝑥) ∈ V
11	10	pwex 5320	. . . . . . . . . . 11 ⊢ 𝒫 (𝑥 × 𝑥) ∈ V
12	11	pwex 5320	. . . . . . . . . 10 ⊢ 𝒫 𝒫 (𝑥 × 𝑥) ∈ V
13	9, 12	eqeltri 2829	. . . . . . . . 9 ⊢ {𝑢 ∣ 𝑢 ∈ 𝒫 𝒫 (𝑥 × 𝑥)} ∈ V
14	8, 13	eqeltrri 2830	. . . . . . . 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 4015	. . . . . . . 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 5262	. . . . . . 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 24117	. . . . . . . 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 6946	. . . . . . 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 4015	. . . . . 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 3977	. . . . 5 ⊢ (UnifOn‘𝑥) ⊆ {𝑢 ∣ (𝑥 × 𝑥) ∈ 𝑢}
24	23	sseli 3926	. . . 4 ⊢ (∅ ∈ (UnifOn‘𝑥) → ∅ ∈ {𝑢 ∣ (𝑥 × 𝑥) ∈ 𝑢})
25	5, 24	mto 197	. . 3 ⊢ ¬ ∅ ∈ (UnifOn‘𝑥)
26	25	nex 1801	. 2 ⊢ ¬ ∃𝑥∅ ∈ (UnifOn‘𝑥)
27	18	funmpt2 6525	. . . 4 ⊢ Fun UnifOn
28		elunirn 7191	. . . 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 24118	. . . . 5 ⊢ UnifOn Fn V
31		fndm 6589	. . . . 5 ⊢ (UnifOn Fn V → dom UnifOn = V)
32	30, 31	ax-mp 5	. . . 4 ⊢ dom UnifOn = V
33	32	rexeqi 3292	. . 3 ⊢ (∃𝑥 ∈ dom UnifOn∅ ∈ (UnifOn‘𝑥) ↔ ∃𝑥 ∈ V ∅ ∈ (UnifOn‘𝑥))
34		rexv 3465	. . 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 1541  ∃wex 1780   ∈ wcel 2113  {cab 2711  ∀wral 3048  ∃wrex 3057  Vcvv 3437   ∩ cin 3897   ⊆ wss 3898  ∅c0 4282  𝒫 cpw 4549  ∪ cuni 4858   I cid 5513   × cxp 5617  ^◡ccnv 5618  dom cdm 5619  ran crn 5620   ↾ cres 5621   ∘ ccom 5623  Fun wfun 6480   Fn wfn 6481  ‘cfv 6486  UnifOncust 24116

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-fv 6494  df-ust 24117

This theorem is referenced by: (None)