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Theorem ustn0 22237
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 4120 . . . . 5 ¬ (𝑥 × 𝑥) ∈ ∅
2 0ex 4984 . . . . . 6 ∅ ∈ V
3 eleq2 2874 . . . . . 6 (𝑢 = ∅ → ((𝑥 × 𝑥) ∈ 𝑢 ↔ (𝑥 × 𝑥) ∈ ∅))
42, 3elab 3545 . . . . 5 (∅ ∈ {𝑢 ∣ (𝑥 × 𝑥) ∈ 𝑢} ↔ (𝑥 × 𝑥) ∈ ∅)
51, 4mtbir 314 . . . 4 ¬ ∅ ∈ {𝑢 ∣ (𝑥 × 𝑥) ∈ 𝑢}
6 vex 3394 . . . . . . 7 𝑥 ∈ V
7 selpw 4358 . . . . . . . . . 10 (𝑢 ∈ 𝒫 𝒫 (𝑥 × 𝑥) ↔ 𝑢 ⊆ 𝒫 (𝑥 × 𝑥))
87abbii 2923 . . . . . . . . 9 {𝑢𝑢 ∈ 𝒫 𝒫 (𝑥 × 𝑥)} = {𝑢𝑢 ⊆ 𝒫 (𝑥 × 𝑥)}
9 abid2 2929 . . . . . . . . . 10 {𝑢𝑢 ∈ 𝒫 𝒫 (𝑥 × 𝑥)} = 𝒫 𝒫 (𝑥 × 𝑥)
106, 6xpex 7192 . . . . . . . . . . . 12 (𝑥 × 𝑥) ∈ V
1110pwex 5050 . . . . . . . . . . 11 𝒫 (𝑥 × 𝑥) ∈ V
1211pwex 5050 . . . . . . . . . 10 𝒫 𝒫 (𝑥 × 𝑥) ∈ V
139, 12eqeltri 2881 . . . . . . . . 9 {𝑢𝑢 ∈ 𝒫 𝒫 (𝑥 × 𝑥)} ∈ V
148, 13eqeltrri 2882 . . . . . . . 8 {𝑢𝑢 ⊆ 𝒫 (𝑥 × 𝑥)} ∈ V
15 simp1 1159 . . . . . . . . 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 3871 . . . . . . . 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 4998 . . . . . . 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 22217 . . . . . . . 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 6512 . . . . . . 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 675 . . . . . 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 1160 . . . . . . 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 3871 . . . . . 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 3832 . . . . 5 (UnifOn‘𝑥) ⊆ {𝑢 ∣ (𝑥 × 𝑥) ∈ 𝑢}
2423sseli 3794 . . . 4 (∅ ∈ (UnifOn‘𝑥) → ∅ ∈ {𝑢 ∣ (𝑥 × 𝑥) ∈ 𝑢})
255, 24mto 188 . . 3 ¬ ∅ ∈ (UnifOn‘𝑥)
2625nex 1882 . 2 ¬ ∃𝑥∅ ∈ (UnifOn‘𝑥)
2718funmpt2 6140 . . . 4 Fun UnifOn
28 elunirn 6733 . . . 4 (Fun UnifOn → (∅ ∈ ran UnifOn ↔ ∃𝑥 ∈ dom UnifOn∅ ∈ (UnifOn‘𝑥)))
2927, 28ax-mp 5 . . 3 (∅ ∈ ran UnifOn ↔ ∃𝑥 ∈ dom UnifOn∅ ∈ (UnifOn‘𝑥))
30 ustfn 22218 . . . . 5 UnifOn Fn V
31 fndm 6201 . . . . 5 (UnifOn Fn V → dom UnifOn = V)
3230, 31ax-mp 5 . . . 4 dom UnifOn = V
3332rexeqi 3332 . . 3 (∃𝑥 ∈ dom UnifOn∅ ∈ (UnifOn‘𝑥) ↔ ∃𝑥 ∈ V ∅ ∈ (UnifOn‘𝑥))
34 rexv 3414 . . 3 (∃𝑥 ∈ V ∅ ∈ (UnifOn‘𝑥) ↔ ∃𝑥∅ ∈ (UnifOn‘𝑥))
3529, 33, 343bitri 288 . 2 (∅ ∈ ran UnifOn ↔ ∃𝑥∅ ∈ (UnifOn‘𝑥))
3626, 35mtbir 314 1 ¬ ∅ ∈ ran UnifOn
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  w3a 1100   = wceq 1637  wex 1859  wcel 2156  {cab 2792  wral 3096  wrex 3097  Vcvv 3391  cin 3768  wss 3769  c0 4116  𝒫 cpw 4351   cuni 4630   I cid 5218   × cxp 5309  ccnv 5310  dom cdm 5311  ran crn 5312  cres 5313  ccom 5315  Fun wfun 6095   Fn wfn 6096  cfv 6101  UnifOncust 22216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6064  df-fun 6103  df-fn 6104  df-fv 6109  df-ust 22217
This theorem is referenced by: (None)
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