Theorem ustn0 24216

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 4333	. . . . 5 ⊢ ¬ (𝑥 × 𝑥) ∈ ∅
2		0ex 5312	. . . . . 6 ⊢ ∅ ∈ V
3		eleq2 2815	. . . . . 6 ⊢ (𝑢 = ∅ → ((𝑥 × 𝑥) ∈ 𝑢 ↔ (𝑥 × 𝑥) ∈ ∅))
4	2, 3	elab 3666	. . . . 5 ⊢ (∅ ∈ {𝑢 ∣ (𝑥 × 𝑥) ∈ 𝑢} ↔ (𝑥 × 𝑥) ∈ ∅)
5	1, 4	mtbir 322	. . . 4 ⊢ ¬ ∅ ∈ {𝑢 ∣ (𝑥 × 𝑥) ∈ 𝑢}
6		vex 3466	. . . . . . 7 ⊢ 𝑥 ∈ V
7		velpw 4612	. . . . . . . . . 10 ⊢ (𝑢 ∈ 𝒫 𝒫 (𝑥 × 𝑥) ↔ 𝑢 ⊆ 𝒫 (𝑥 × 𝑥))
8	7	abbii 2796	. . . . . . . . 9 ⊢ {𝑢 ∣ 𝑢 ∈ 𝒫 𝒫 (𝑥 × 𝑥)} = {𝑢 ∣ 𝑢 ⊆ 𝒫 (𝑥 × 𝑥)}
9		abid2 2864	. . . . . . . . . 10 ⊢ {𝑢 ∣ 𝑢 ∈ 𝒫 𝒫 (𝑥 × 𝑥)} = 𝒫 𝒫 (𝑥 × 𝑥)
10	6, 6	xpex 7761	. . . . . . . . . . . 12 ⊢ (𝑥 × 𝑥) ∈ V
11	10	pwex 5384	. . . . . . . . . . 11 ⊢ 𝒫 (𝑥 × 𝑥) ∈ V
12	11	pwex 5384	. . . . . . . . . 10 ⊢ 𝒫 𝒫 (𝑥 × 𝑥) ∈ V
13	9, 12	eqeltri 2822	. . . . . . . . 9 ⊢ {𝑢 ∣ 𝑢 ∈ 𝒫 𝒫 (𝑥 × 𝑥)} ∈ V
14	8, 13	eqeltrri 2823	. . . . . . . 8 ⊢ {𝑢 ∣ 𝑢 ⊆ 𝒫 (𝑥 × 𝑥)} ∈ V
15		simp1 1133	. . . . . . . . 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 4062	. . . . . . . 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 5327	. . . . . . 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 24196	. . . . . . . 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 7020	. . . . . . 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 690	. . . . . 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 1134	. . . . . . 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 4062	. . . . . 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 4014	. . . . 5 ⊢ (UnifOn‘𝑥) ⊆ {𝑢 ∣ (𝑥 × 𝑥) ∈ 𝑢}
24	23	sseli 3975	. . . 4 ⊢ (∅ ∈ (UnifOn‘𝑥) → ∅ ∈ {𝑢 ∣ (𝑥 × 𝑥) ∈ 𝑢})
25	5, 24	mto 196	. . 3 ⊢ ¬ ∅ ∈ (UnifOn‘𝑥)
26	25	nex 1795	. 2 ⊢ ¬ ∃𝑥∅ ∈ (UnifOn‘𝑥)
27	18	funmpt2 6598	. . . 4 ⊢ Fun UnifOn
28		elunirn 7266	. . . 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 24197	. . . . 5 ⊢ UnifOn Fn V
31		fndm 6663	. . . . 5 ⊢ (UnifOn Fn V → dom UnifOn = V)
32	30, 31	ax-mp 5	. . . 4 ⊢ dom UnifOn = V
33	32	rexeqi 3314	. . 3 ⊢ (∃𝑥 ∈ dom UnifOn∅ ∈ (UnifOn‘𝑥) ↔ ∃𝑥 ∈ V ∅ ∈ (UnifOn‘𝑥))
34		rexv 3490	. . 3 ⊢ (∃𝑥 ∈ V ∅ ∈ (UnifOn‘𝑥) ↔ ∃𝑥∅ ∈ (UnifOn‘𝑥))
35	29, 33, 34	3bitri 296	. 2 ⊢ (∅ ∈ ∪ ran UnifOn ↔ ∃𝑥∅ ∈ (UnifOn‘𝑥))
36	26, 35	mtbir 322	1 ⊢ ¬ ∅ ∈ ∪ ran UnifOn

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ w3a 1084   = wceq 1534  ∃wex 1774   ∈ wcel 2099  {cab 2703  ∀wral 3051  ∃wrex 3060  Vcvv 3462   ∩ cin 3946   ⊆ wss 3947  ∅c0 4325  𝒫 cpw 4607  ∪ cuni 4913   I cid 5579   × cxp 5680  ^◡ccnv 5681  dom cdm 5682  ran crn 5683   ↾ cres 5684   ∘ ccom 5686  Fun wfun 6548   Fn wfn 6549  ‘cfv 6554  UnifOncust 24195

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-fv 6562  df-ust 24196

This theorem is referenced by: (None)