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Theorem ust0 22302
Description: The unique uniform structure of the empty set is the empty set. Remark 3 of [BourbakiTop1] p. II.2. (Contributed by Thierry Arnoux, 15-Nov-2017.)
Assertion
Ref Expression
ust0 (UnifOn‘∅) = {{∅}}

Proof of Theorem ust0
Dummy variables 𝑣 𝑢 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0ex 4950 . . . . . . . 8 ∅ ∈ V
2 isust 22286 . . . . . . . 8 (∅ ∈ V → (𝑢 ∈ (UnifOn‘∅) ↔ (𝑢 ⊆ 𝒫 (∅ × ∅) ∧ (∅ × ∅) ∈ 𝑢 ∧ ∀𝑣𝑢 (∀𝑤 ∈ 𝒫 (∅ × ∅)(𝑣𝑤𝑤𝑢) ∧ ∀𝑤𝑢 (𝑣𝑤) ∈ 𝑢 ∧ (( I ↾ ∅) ⊆ 𝑣𝑣𝑢 ∧ ∃𝑤𝑢 (𝑤𝑤) ⊆ 𝑣)))))
31, 2ax-mp 5 . . . . . . 7 (𝑢 ∈ (UnifOn‘∅) ↔ (𝑢 ⊆ 𝒫 (∅ × ∅) ∧ (∅ × ∅) ∈ 𝑢 ∧ ∀𝑣𝑢 (∀𝑤 ∈ 𝒫 (∅ × ∅)(𝑣𝑤𝑤𝑢) ∧ ∀𝑤𝑢 (𝑣𝑤) ∈ 𝑢 ∧ (( I ↾ ∅) ⊆ 𝑣𝑣𝑢 ∧ ∃𝑤𝑢 (𝑤𝑤) ⊆ 𝑣))))
43simp1bi 1175 . . . . . 6 (𝑢 ∈ (UnifOn‘∅) → 𝑢 ⊆ 𝒫 (∅ × ∅))
5 0xp 5369 . . . . . . . 8 (∅ × ∅) = ∅
65pweqi 4319 . . . . . . 7 𝒫 (∅ × ∅) = 𝒫 ∅
7 pw0 4497 . . . . . . 7 𝒫 ∅ = {∅}
86, 7eqtri 2787 . . . . . 6 𝒫 (∅ × ∅) = {∅}
94, 8syl6sseq 3811 . . . . 5 (𝑢 ∈ (UnifOn‘∅) → 𝑢 ⊆ {∅})
10 ustbasel 22289 . . . . . . 7 (𝑢 ∈ (UnifOn‘∅) → (∅ × ∅) ∈ 𝑢)
115, 10syl5eqelr 2849 . . . . . 6 (𝑢 ∈ (UnifOn‘∅) → ∅ ∈ 𝑢)
1211snssd 4494 . . . . 5 (𝑢 ∈ (UnifOn‘∅) → {∅} ⊆ 𝑢)
139, 12eqssd 3778 . . . 4 (𝑢 ∈ (UnifOn‘∅) → 𝑢 = {∅})
14 velsn 4350 . . . 4 (𝑢 ∈ {{∅}} ↔ 𝑢 = {∅})
1513, 14sylibr 225 . . 3 (𝑢 ∈ (UnifOn‘∅) → 𝑢 ∈ {{∅}})
1615ssriv 3765 . 2 (UnifOn‘∅) ⊆ {{∅}}
178eqimss2i 3820 . . . 4 {∅} ⊆ 𝒫 (∅ × ∅)
181snid 4366 . . . . 5 ∅ ∈ {∅}
195, 18eqeltri 2840 . . . 4 (∅ × ∅) ∈ {∅}
2018a1i 11 . . . . . 6 (∅ ⊆ ∅ → ∅ ∈ {∅})
218raleqi 3290 . . . . . . 7 (∀𝑤 ∈ 𝒫 (∅ × ∅)(∅ ⊆ 𝑤𝑤 ∈ {∅}) ↔ ∀𝑤 ∈ {∅} (∅ ⊆ 𝑤𝑤 ∈ {∅}))
22 sseq2 3787 . . . . . . . . 9 (𝑤 = ∅ → (∅ ⊆ 𝑤 ↔ ∅ ⊆ ∅))
23 eleq1 2832 . . . . . . . . 9 (𝑤 = ∅ → (𝑤 ∈ {∅} ↔ ∅ ∈ {∅}))
2422, 23imbi12d 335 . . . . . . . 8 (𝑤 = ∅ → ((∅ ⊆ 𝑤𝑤 ∈ {∅}) ↔ (∅ ⊆ ∅ → ∅ ∈ {∅})))
251, 24ralsn 4379 . . . . . . 7 (∀𝑤 ∈ {∅} (∅ ⊆ 𝑤𝑤 ∈ {∅}) ↔ (∅ ⊆ ∅ → ∅ ∈ {∅}))
2621, 25bitri 266 . . . . . 6 (∀𝑤 ∈ 𝒫 (∅ × ∅)(∅ ⊆ 𝑤𝑤 ∈ {∅}) ↔ (∅ ⊆ ∅ → ∅ ∈ {∅}))
2720, 26mpbir 222 . . . . 5 𝑤 ∈ 𝒫 (∅ × ∅)(∅ ⊆ 𝑤𝑤 ∈ {∅})
28 inidm 3982 . . . . . . 7 (∅ ∩ ∅) = ∅
2928, 18eqeltri 2840 . . . . . 6 (∅ ∩ ∅) ∈ {∅}
30 ineq2 3970 . . . . . . . 8 (𝑤 = ∅ → (∅ ∩ 𝑤) = (∅ ∩ ∅))
3130eleq1d 2829 . . . . . . 7 (𝑤 = ∅ → ((∅ ∩ 𝑤) ∈ {∅} ↔ (∅ ∩ ∅) ∈ {∅}))
321, 31ralsn 4379 . . . . . 6 (∀𝑤 ∈ {∅} (∅ ∩ 𝑤) ∈ {∅} ↔ (∅ ∩ ∅) ∈ {∅})
3329, 32mpbir 222 . . . . 5 𝑤 ∈ {∅} (∅ ∩ 𝑤) ∈ {∅}
34 res0 5569 . . . . . . 7 ( I ↾ ∅) = ∅
3534eqimssi 3819 . . . . . 6 ( I ↾ ∅) ⊆ ∅
36 cnv0 5718 . . . . . . 7 ∅ = ∅
3736, 18eqeltri 2840 . . . . . 6 ∅ ∈ {∅}
38 0trrel 14009 . . . . . . 7 (∅ ∘ ∅) ⊆ ∅
39 id 22 . . . . . . . . . 10 (𝑤 = ∅ → 𝑤 = ∅)
4039, 39coeq12d 5455 . . . . . . . . 9 (𝑤 = ∅ → (𝑤𝑤) = (∅ ∘ ∅))
4140sseq1d 3792 . . . . . . . 8 (𝑤 = ∅ → ((𝑤𝑤) ⊆ ∅ ↔ (∅ ∘ ∅) ⊆ ∅))
421, 41rexsn 4380 . . . . . . 7 (∃𝑤 ∈ {∅} (𝑤𝑤) ⊆ ∅ ↔ (∅ ∘ ∅) ⊆ ∅)
4338, 42mpbir 222 . . . . . 6 𝑤 ∈ {∅} (𝑤𝑤) ⊆ ∅
4435, 37, 433pm3.2i 1438 . . . . 5 (( I ↾ ∅) ⊆ ∅ ∧ ∅ ∈ {∅} ∧ ∃𝑤 ∈ {∅} (𝑤𝑤) ⊆ ∅)
45 sseq1 3786 . . . . . . . . 9 (𝑣 = ∅ → (𝑣𝑤 ↔ ∅ ⊆ 𝑤))
4645imbi1d 332 . . . . . . . 8 (𝑣 = ∅ → ((𝑣𝑤𝑤 ∈ {∅}) ↔ (∅ ⊆ 𝑤𝑤 ∈ {∅})))
4746ralbidv 3133 . . . . . . 7 (𝑣 = ∅ → (∀𝑤 ∈ 𝒫 (∅ × ∅)(𝑣𝑤𝑤 ∈ {∅}) ↔ ∀𝑤 ∈ 𝒫 (∅ × ∅)(∅ ⊆ 𝑤𝑤 ∈ {∅})))
48 ineq1 3969 . . . . . . . . 9 (𝑣 = ∅ → (𝑣𝑤) = (∅ ∩ 𝑤))
4948eleq1d 2829 . . . . . . . 8 (𝑣 = ∅ → ((𝑣𝑤) ∈ {∅} ↔ (∅ ∩ 𝑤) ∈ {∅}))
5049ralbidv 3133 . . . . . . 7 (𝑣 = ∅ → (∀𝑤 ∈ {∅} (𝑣𝑤) ∈ {∅} ↔ ∀𝑤 ∈ {∅} (∅ ∩ 𝑤) ∈ {∅}))
51 sseq2 3787 . . . . . . . 8 (𝑣 = ∅ → (( I ↾ ∅) ⊆ 𝑣 ↔ ( I ↾ ∅) ⊆ ∅))
52 cnveq 5464 . . . . . . . . 9 (𝑣 = ∅ → 𝑣 = ∅)
5352eleq1d 2829 . . . . . . . 8 (𝑣 = ∅ → (𝑣 ∈ {∅} ↔ ∅ ∈ {∅}))
54 sseq2 3787 . . . . . . . . 9 (𝑣 = ∅ → ((𝑤𝑤) ⊆ 𝑣 ↔ (𝑤𝑤) ⊆ ∅))
5554rexbidv 3199 . . . . . . . 8 (𝑣 = ∅ → (∃𝑤 ∈ {∅} (𝑤𝑤) ⊆ 𝑣 ↔ ∃𝑤 ∈ {∅} (𝑤𝑤) ⊆ ∅))
5651, 53, 553anbi123d 1560 . . . . . . 7 (𝑣 = ∅ → ((( I ↾ ∅) ⊆ 𝑣𝑣 ∈ {∅} ∧ ∃𝑤 ∈ {∅} (𝑤𝑤) ⊆ 𝑣) ↔ (( I ↾ ∅) ⊆ ∅ ∧ ∅ ∈ {∅} ∧ ∃𝑤 ∈ {∅} (𝑤𝑤) ⊆ ∅)))
5747, 50, 563anbi123d 1560 . . . . . 6 (𝑣 = ∅ → ((∀𝑤 ∈ 𝒫 (∅ × ∅)(𝑣𝑤𝑤 ∈ {∅}) ∧ ∀𝑤 ∈ {∅} (𝑣𝑤) ∈ {∅} ∧ (( I ↾ ∅) ⊆ 𝑣𝑣 ∈ {∅} ∧ ∃𝑤 ∈ {∅} (𝑤𝑤) ⊆ 𝑣)) ↔ (∀𝑤 ∈ 𝒫 (∅ × ∅)(∅ ⊆ 𝑤𝑤 ∈ {∅}) ∧ ∀𝑤 ∈ {∅} (∅ ∩ 𝑤) ∈ {∅} ∧ (( I ↾ ∅) ⊆ ∅ ∧ ∅ ∈ {∅} ∧ ∃𝑤 ∈ {∅} (𝑤𝑤) ⊆ ∅))))
581, 57ralsn 4379 . . . . 5 (∀𝑣 ∈ {∅} (∀𝑤 ∈ 𝒫 (∅ × ∅)(𝑣𝑤𝑤 ∈ {∅}) ∧ ∀𝑤 ∈ {∅} (𝑣𝑤) ∈ {∅} ∧ (( I ↾ ∅) ⊆ 𝑣𝑣 ∈ {∅} ∧ ∃𝑤 ∈ {∅} (𝑤𝑤) ⊆ 𝑣)) ↔ (∀𝑤 ∈ 𝒫 (∅ × ∅)(∅ ⊆ 𝑤𝑤 ∈ {∅}) ∧ ∀𝑤 ∈ {∅} (∅ ∩ 𝑤) ∈ {∅} ∧ (( I ↾ ∅) ⊆ ∅ ∧ ∅ ∈ {∅} ∧ ∃𝑤 ∈ {∅} (𝑤𝑤) ⊆ ∅)))
5927, 33, 44, 58mpbir3an 1441 . . . 4 𝑣 ∈ {∅} (∀𝑤 ∈ 𝒫 (∅ × ∅)(𝑣𝑤𝑤 ∈ {∅}) ∧ ∀𝑤 ∈ {∅} (𝑣𝑤) ∈ {∅} ∧ (( I ↾ ∅) ⊆ 𝑣𝑣 ∈ {∅} ∧ ∃𝑤 ∈ {∅} (𝑤𝑤) ⊆ 𝑣))
60 isust 22286 . . . . 5 (∅ ∈ V → ({∅} ∈ (UnifOn‘∅) ↔ ({∅} ⊆ 𝒫 (∅ × ∅) ∧ (∅ × ∅) ∈ {∅} ∧ ∀𝑣 ∈ {∅} (∀𝑤 ∈ 𝒫 (∅ × ∅)(𝑣𝑤𝑤 ∈ {∅}) ∧ ∀𝑤 ∈ {∅} (𝑣𝑤) ∈ {∅} ∧ (( I ↾ ∅) ⊆ 𝑣𝑣 ∈ {∅} ∧ ∃𝑤 ∈ {∅} (𝑤𝑤) ⊆ 𝑣)))))
611, 60ax-mp 5 . . . 4 ({∅} ∈ (UnifOn‘∅) ↔ ({∅} ⊆ 𝒫 (∅ × ∅) ∧ (∅ × ∅) ∈ {∅} ∧ ∀𝑣 ∈ {∅} (∀𝑤 ∈ 𝒫 (∅ × ∅)(𝑣𝑤𝑤 ∈ {∅}) ∧ ∀𝑤 ∈ {∅} (𝑣𝑤) ∈ {∅} ∧ (( I ↾ ∅) ⊆ 𝑣𝑣 ∈ {∅} ∧ ∃𝑤 ∈ {∅} (𝑤𝑤) ⊆ 𝑣))))
6217, 19, 59, 61mpbir3an 1441 . . 3 {∅} ∈ (UnifOn‘∅)
63 snssi 4493 . . 3 ({∅} ∈ (UnifOn‘∅) → {{∅}} ⊆ (UnifOn‘∅))
6462, 63ax-mp 5 . 2 {{∅}} ⊆ (UnifOn‘∅)
6516, 64eqssi 3777 1 (UnifOn‘∅) = {{∅}}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  w3a 1107   = wceq 1652  wcel 2155  wral 3055  wrex 3056  Vcvv 3350  cin 3731  wss 3732  c0 4079  𝒫 cpw 4315  {csn 4334   I cid 5184   × cxp 5275  ccnv 5276  cres 5279  ccom 5281  cfv 6068  UnifOncust 22282
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-res 5289  df-iota 6031  df-fun 6070  df-fv 6076  df-ust 22283
This theorem is referenced by:  isusp  22344
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