MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ustn0 Structured version   Visualization version   GIF version

Theorem ustn0 21781
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.
StepHypRef Expression
1 noel 3877 . . . . 5 ¬ (𝑥 × 𝑥) ∈ ∅
2 0ex 4712 . . . . . 6 ∅ ∈ V
3 eleq2 2676 . . . . . 6 (𝑢 = ∅ → ((𝑥 × 𝑥) ∈ 𝑢 ↔ (𝑥 × 𝑥) ∈ ∅))
42, 3elab 3318 . . . . 5 (∅ ∈ {𝑢 ∣ (𝑥 × 𝑥) ∈ 𝑢} ↔ (𝑥 × 𝑥) ∈ ∅)
51, 4mtbir 311 . . . 4 ¬ ∅ ∈ {𝑢 ∣ (𝑥 × 𝑥) ∈ 𝑢}
6 vex 3175 . . . . . . 7 𝑥 ∈ V
7 selpw 4114 . . . . . . . . . 10 (𝑢 ∈ 𝒫 𝒫 (𝑥 × 𝑥) ↔ 𝑢 ⊆ 𝒫 (𝑥 × 𝑥))
87abbii 2725 . . . . . . . . 9 {𝑢𝑢 ∈ 𝒫 𝒫 (𝑥 × 𝑥)} = {𝑢𝑢 ⊆ 𝒫 (𝑥 × 𝑥)}
9 abid2 2731 . . . . . . . . . 10 {𝑢𝑢 ∈ 𝒫 𝒫 (𝑥 × 𝑥)} = 𝒫 𝒫 (𝑥 × 𝑥)
106, 6xpex 6837 . . . . . . . . . . . 12 (𝑥 × 𝑥) ∈ V
1110pwex 4768 . . . . . . . . . . 11 𝒫 (𝑥 × 𝑥) ∈ V
1211pwex 4768 . . . . . . . . . 10 𝒫 𝒫 (𝑥 × 𝑥) ∈ V
139, 12eqeltri 2683 . . . . . . . . 9 {𝑢𝑢 ∈ 𝒫 𝒫 (𝑥 × 𝑥)} ∈ V
148, 13eqeltrri 2684 . . . . . . . 8 {𝑢𝑢 ⊆ 𝒫 (𝑥 × 𝑥)} ∈ V
15 simp1 1053 . . . . . . . . 9 ((𝑢 ⊆ 𝒫 (𝑥 × 𝑥) ∧ (𝑥 × 𝑥) ∈ 𝑢 ∧ ∀𝑣𝑢 (∀𝑤 ∈ 𝒫 (𝑥 × 𝑥)(𝑣𝑤𝑤𝑢) ∧ ∀𝑤𝑢 (𝑣𝑤) ∈ 𝑢 ∧ (( I ↾ 𝑥) ⊆ 𝑣𝑣𝑢 ∧ ∃𝑤𝑢 (𝑤𝑤) ⊆ 𝑣))) → 𝑢 ⊆ 𝒫 (𝑥 × 𝑥))
1615ss2abi 3636 . . . . . . . 8 {𝑢 ∣ (𝑢 ⊆ 𝒫 (𝑥 × 𝑥) ∧ (𝑥 × 𝑥) ∈ 𝑢 ∧ ∀𝑣𝑢 (∀𝑤 ∈ 𝒫 (𝑥 × 𝑥)(𝑣𝑤𝑤𝑢) ∧ ∀𝑤𝑢 (𝑣𝑤) ∈ 𝑢 ∧ (( I ↾ 𝑥) ⊆ 𝑣𝑣𝑢 ∧ ∃𝑤𝑢 (𝑤𝑤) ⊆ 𝑣)))} ⊆ {𝑢𝑢 ⊆ 𝒫 (𝑥 × 𝑥)}
1714, 16ssexi 4725 . . . . . . 7 {𝑢 ∣ (𝑢 ⊆ 𝒫 (𝑥 × 𝑥) ∧ (𝑥 × 𝑥) ∈ 𝑢 ∧ ∀𝑣𝑢 (∀𝑤 ∈ 𝒫 (𝑥 × 𝑥)(𝑣𝑤𝑤𝑢) ∧ ∀𝑤𝑢 (𝑣𝑤) ∈ 𝑢 ∧ (( I ↾ 𝑥) ⊆ 𝑣𝑣𝑢 ∧ ∃𝑤𝑢 (𝑤𝑤) ⊆ 𝑣)))} ∈ V
18 df-ust 21761 . . . . . . . 8 UnifOn = (𝑥 ∈ V ↦ {𝑢 ∣ (𝑢 ⊆ 𝒫 (𝑥 × 𝑥) ∧ (𝑥 × 𝑥) ∈ 𝑢 ∧ ∀𝑣𝑢 (∀𝑤 ∈ 𝒫 (𝑥 × 𝑥)(𝑣𝑤𝑤𝑢) ∧ ∀𝑤𝑢 (𝑣𝑤) ∈ 𝑢 ∧ (( I ↾ 𝑥) ⊆ 𝑣𝑣𝑢 ∧ ∃𝑤𝑢 (𝑤𝑤) ⊆ 𝑣)))})
1918fvmpt2 6184 . . . . . . 7 ((𝑥 ∈ V ∧ {𝑢 ∣ (𝑢 ⊆ 𝒫 (𝑥 × 𝑥) ∧ (𝑥 × 𝑥) ∈ 𝑢 ∧ ∀𝑣𝑢 (∀𝑤 ∈ 𝒫 (𝑥 × 𝑥)(𝑣𝑤𝑤𝑢) ∧ ∀𝑤𝑢 (𝑣𝑤) ∈ 𝑢 ∧ (( I ↾ 𝑥) ⊆ 𝑣𝑣𝑢 ∧ ∃𝑤𝑢 (𝑤𝑤) ⊆ 𝑣)))} ∈ V) → (UnifOn‘𝑥) = {𝑢 ∣ (𝑢 ⊆ 𝒫 (𝑥 × 𝑥) ∧ (𝑥 × 𝑥) ∈ 𝑢 ∧ ∀𝑣𝑢 (∀𝑤 ∈ 𝒫 (𝑥 × 𝑥)(𝑣𝑤𝑤𝑢) ∧ ∀𝑤𝑢 (𝑣𝑤) ∈ 𝑢 ∧ (( I ↾ 𝑥) ⊆ 𝑣𝑣𝑢 ∧ ∃𝑤𝑢 (𝑤𝑤) ⊆ 𝑣)))})
206, 17, 19mp2an 703 . . . . . 6 (UnifOn‘𝑥) = {𝑢 ∣ (𝑢 ⊆ 𝒫 (𝑥 × 𝑥) ∧ (𝑥 × 𝑥) ∈ 𝑢 ∧ ∀𝑣𝑢 (∀𝑤 ∈ 𝒫 (𝑥 × 𝑥)(𝑣𝑤𝑤𝑢) ∧ ∀𝑤𝑢 (𝑣𝑤) ∈ 𝑢 ∧ (( I ↾ 𝑥) ⊆ 𝑣𝑣𝑢 ∧ ∃𝑤𝑢 (𝑤𝑤) ⊆ 𝑣)))}
21 simp2 1054 . . . . . . 7 ((𝑢 ⊆ 𝒫 (𝑥 × 𝑥) ∧ (𝑥 × 𝑥) ∈ 𝑢 ∧ ∀𝑣𝑢 (∀𝑤 ∈ 𝒫 (𝑥 × 𝑥)(𝑣𝑤𝑤𝑢) ∧ ∀𝑤𝑢 (𝑣𝑤) ∈ 𝑢 ∧ (( I ↾ 𝑥) ⊆ 𝑣𝑣𝑢 ∧ ∃𝑤𝑢 (𝑤𝑤) ⊆ 𝑣))) → (𝑥 × 𝑥) ∈ 𝑢)
2221ss2abi 3636 . . . . . 6 {𝑢 ∣ (𝑢 ⊆ 𝒫 (𝑥 × 𝑥) ∧ (𝑥 × 𝑥) ∈ 𝑢 ∧ ∀𝑣𝑢 (∀𝑤 ∈ 𝒫 (𝑥 × 𝑥)(𝑣𝑤𝑤𝑢) ∧ ∀𝑤𝑢 (𝑣𝑤) ∈ 𝑢 ∧ (( I ↾ 𝑥) ⊆ 𝑣𝑣𝑢 ∧ ∃𝑤𝑢 (𝑤𝑤) ⊆ 𝑣)))} ⊆ {𝑢 ∣ (𝑥 × 𝑥) ∈ 𝑢}
2320, 22eqsstri 3597 . . . . 5 (UnifOn‘𝑥) ⊆ {𝑢 ∣ (𝑥 × 𝑥) ∈ 𝑢}
2423sseli 3563 . . . 4 (∅ ∈ (UnifOn‘𝑥) → ∅ ∈ {𝑢 ∣ (𝑥 × 𝑥) ∈ 𝑢})
255, 24mto 186 . . 3 ¬ ∅ ∈ (UnifOn‘𝑥)
2625nex 1721 . 2 ¬ ∃𝑥∅ ∈ (UnifOn‘𝑥)
2718funmpt2 5826 . . . 4 Fun UnifOn
28 elunirn 6390 . . . 4 (Fun UnifOn → (∅ ∈ ran UnifOn ↔ ∃𝑥 ∈ dom UnifOn∅ ∈ (UnifOn‘𝑥)))
2927, 28ax-mp 5 . . 3 (∅ ∈ ran UnifOn ↔ ∃𝑥 ∈ dom UnifOn∅ ∈ (UnifOn‘𝑥))
30 ustfn 21762 . . . . 5 UnifOn Fn V
31 fndm 5889 . . . . 5 (UnifOn Fn V → dom UnifOn = V)
3230, 31ax-mp 5 . . . 4 dom UnifOn = V
3332rexeqi 3119 . . 3 (∃𝑥 ∈ dom UnifOn∅ ∈ (UnifOn‘𝑥) ↔ ∃𝑥 ∈ V ∅ ∈ (UnifOn‘𝑥))
34 rexv 3192 . . 3 (∃𝑥 ∈ V ∅ ∈ (UnifOn‘𝑥) ↔ ∃𝑥∅ ∈ (UnifOn‘𝑥))
3529, 33, 343bitri 284 . 2 (∅ ∈ ran UnifOn ↔ ∃𝑥∅ ∈ (UnifOn‘𝑥))
3626, 35mtbir 311 1 ¬ ∅ ∈ ran UnifOn
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  w3a 1030   = wceq 1474  wex 1694  wcel 1976  {cab 2595  wral 2895  wrex 2896  Vcvv 3172  cin 3538  wss 3539  c0 3873  𝒫 cpw 4107   cuni 4366   I cid 4937   × cxp 5025  ccnv 5026  dom cdm 5027  ran crn 5028  cres 5029  ccom 5031  Fun wfun 5783   Fn wfn 5784  cfv 5789  UnifOncust 21760
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pow 4763  ax-pr 4827  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4942  df-xp 5033  df-rel 5034  df-cnv 5035  df-co 5036  df-dm 5037  df-rn 5038  df-res 5039  df-ima 5040  df-iota 5753  df-fun 5791  df-fn 5792  df-fv 5797  df-ust 21761
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator