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Theorem ust0 22964
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 5172 . . . . . . . 8 ∅ ∈ V
2 isust 22948 . . . . . . . 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 1146 . . . . . 6 (𝑢 ∈ (UnifOn‘∅) → 𝑢 ⊆ 𝒫 (∅ × ∅))
5 0xp 5614 . . . . . . . 8 (∅ × ∅) = ∅
65pweqi 4503 . . . . . . 7 𝒫 (∅ × ∅) = 𝒫 ∅
7 pw0 4697 . . . . . . 7 𝒫 ∅ = {∅}
86, 7eqtri 2761 . . . . . 6 𝒫 (∅ × ∅) = {∅}
94, 8sseqtrdi 3925 . . . . 5 (𝑢 ∈ (UnifOn‘∅) → 𝑢 ⊆ {∅})
10 ustbasel 22951 . . . . . . 7 (𝑢 ∈ (UnifOn‘∅) → (∅ × ∅) ∈ 𝑢)
115, 10eqeltrrid 2838 . . . . . 6 (𝑢 ∈ (UnifOn‘∅) → ∅ ∈ 𝑢)
1211snssd 4694 . . . . 5 (𝑢 ∈ (UnifOn‘∅) → {∅} ⊆ 𝑢)
139, 12eqssd 3892 . . . 4 (𝑢 ∈ (UnifOn‘∅) → 𝑢 = {∅})
14 velsn 4529 . . . 4 (𝑢 ∈ {{∅}} ↔ 𝑢 = {∅})
1513, 14sylibr 237 . . 3 (𝑢 ∈ (UnifOn‘∅) → 𝑢 ∈ {{∅}})
1615ssriv 3879 . 2 (UnifOn‘∅) ⊆ {{∅}}
178eqimss2i 3934 . . . 4 {∅} ⊆ 𝒫 (∅ × ∅)
181snid 4549 . . . . 5 ∅ ∈ {∅}
195, 18eqeltri 2829 . . . 4 (∅ × ∅) ∈ {∅}
2018a1i 11 . . . . . 6 (∅ ⊆ ∅ → ∅ ∈ {∅})
218raleqi 3313 . . . . . . 7 (∀𝑤 ∈ 𝒫 (∅ × ∅)(∅ ⊆ 𝑤𝑤 ∈ {∅}) ↔ ∀𝑤 ∈ {∅} (∅ ⊆ 𝑤𝑤 ∈ {∅}))
22 sseq2 3901 . . . . . . . . 9 (𝑤 = ∅ → (∅ ⊆ 𝑤 ↔ ∅ ⊆ ∅))
23 eleq1 2820 . . . . . . . . 9 (𝑤 = ∅ → (𝑤 ∈ {∅} ↔ ∅ ∈ {∅}))
2422, 23imbi12d 348 . . . . . . . 8 (𝑤 = ∅ → ((∅ ⊆ 𝑤𝑤 ∈ {∅}) ↔ (∅ ⊆ ∅ → ∅ ∈ {∅})))
251, 24ralsn 4569 . . . . . . 7 (∀𝑤 ∈ {∅} (∅ ⊆ 𝑤𝑤 ∈ {∅}) ↔ (∅ ⊆ ∅ → ∅ ∈ {∅}))
2621, 25bitri 278 . . . . . 6 (∀𝑤 ∈ 𝒫 (∅ × ∅)(∅ ⊆ 𝑤𝑤 ∈ {∅}) ↔ (∅ ⊆ ∅ → ∅ ∈ {∅}))
2720, 26mpbir 234 . . . . 5 𝑤 ∈ 𝒫 (∅ × ∅)(∅ ⊆ 𝑤𝑤 ∈ {∅})
28 inidm 4107 . . . . . . 7 (∅ ∩ ∅) = ∅
2928, 18eqeltri 2829 . . . . . 6 (∅ ∩ ∅) ∈ {∅}
30 ineq2 4095 . . . . . . . 8 (𝑤 = ∅ → (∅ ∩ 𝑤) = (∅ ∩ ∅))
3130eleq1d 2817 . . . . . . 7 (𝑤 = ∅ → ((∅ ∩ 𝑤) ∈ {∅} ↔ (∅ ∩ ∅) ∈ {∅}))
321, 31ralsn 4569 . . . . . 6 (∀𝑤 ∈ {∅} (∅ ∩ 𝑤) ∈ {∅} ↔ (∅ ∩ ∅) ∈ {∅})
3329, 32mpbir 234 . . . . 5 𝑤 ∈ {∅} (∅ ∩ 𝑤) ∈ {∅}
34 res0 5823 . . . . . . 7 ( I ↾ ∅) = ∅
3534eqimssi 3933 . . . . . 6 ( I ↾ ∅) ⊆ ∅
36 cnv0 5967 . . . . . . 7 ∅ = ∅
3736, 18eqeltri 2829 . . . . . 6 ∅ ∈ {∅}
38 0trrel 14423 . . . . . . 7 (∅ ∘ ∅) ⊆ ∅
39 id 22 . . . . . . . . . 10 (𝑤 = ∅ → 𝑤 = ∅)
4039, 39coeq12d 5701 . . . . . . . . 9 (𝑤 = ∅ → (𝑤𝑤) = (∅ ∘ ∅))
4140sseq1d 3906 . . . . . . . 8 (𝑤 = ∅ → ((𝑤𝑤) ⊆ ∅ ↔ (∅ ∘ ∅) ⊆ ∅))
421, 41rexsn 4570 . . . . . . 7 (∃𝑤 ∈ {∅} (𝑤𝑤) ⊆ ∅ ↔ (∅ ∘ ∅) ⊆ ∅)
4338, 42mpbir 234 . . . . . 6 𝑤 ∈ {∅} (𝑤𝑤) ⊆ ∅
4435, 37, 433pm3.2i 1340 . . . . 5 (( I ↾ ∅) ⊆ ∅ ∧ ∅ ∈ {∅} ∧ ∃𝑤 ∈ {∅} (𝑤𝑤) ⊆ ∅)
45 sseq1 3900 . . . . . . . . 9 (𝑣 = ∅ → (𝑣𝑤 ↔ ∅ ⊆ 𝑤))
4645imbi1d 345 . . . . . . . 8 (𝑣 = ∅ → ((𝑣𝑤𝑤 ∈ {∅}) ↔ (∅ ⊆ 𝑤𝑤 ∈ {∅})))
4746ralbidv 3109 . . . . . . 7 (𝑣 = ∅ → (∀𝑤 ∈ 𝒫 (∅ × ∅)(𝑣𝑤𝑤 ∈ {∅}) ↔ ∀𝑤 ∈ 𝒫 (∅ × ∅)(∅ ⊆ 𝑤𝑤 ∈ {∅})))
48 ineq1 4094 . . . . . . . . 9 (𝑣 = ∅ → (𝑣𝑤) = (∅ ∩ 𝑤))
4948eleq1d 2817 . . . . . . . 8 (𝑣 = ∅ → ((𝑣𝑤) ∈ {∅} ↔ (∅ ∩ 𝑤) ∈ {∅}))
5049ralbidv 3109 . . . . . . 7 (𝑣 = ∅ → (∀𝑤 ∈ {∅} (𝑣𝑤) ∈ {∅} ↔ ∀𝑤 ∈ {∅} (∅ ∩ 𝑤) ∈ {∅}))
51 sseq2 3901 . . . . . . . 8 (𝑣 = ∅ → (( I ↾ ∅) ⊆ 𝑣 ↔ ( I ↾ ∅) ⊆ ∅))
52 cnveq 5710 . . . . . . . . 9 (𝑣 = ∅ → 𝑣 = ∅)
5352eleq1d 2817 . . . . . . . 8 (𝑣 = ∅ → (𝑣 ∈ {∅} ↔ ∅ ∈ {∅}))
54 sseq2 3901 . . . . . . . . 9 (𝑣 = ∅ → ((𝑤𝑤) ⊆ 𝑣 ↔ (𝑤𝑤) ⊆ ∅))
5554rexbidv 3206 . . . . . . . 8 (𝑣 = ∅ → (∃𝑤 ∈ {∅} (𝑤𝑤) ⊆ 𝑣 ↔ ∃𝑤 ∈ {∅} (𝑤𝑤) ⊆ ∅))
5651, 53, 553anbi123d 1437 . . . . . . 7 (𝑣 = ∅ → ((( I ↾ ∅) ⊆ 𝑣𝑣 ∈ {∅} ∧ ∃𝑤 ∈ {∅} (𝑤𝑤) ⊆ 𝑣) ↔ (( I ↾ ∅) ⊆ ∅ ∧ ∅ ∈ {∅} ∧ ∃𝑤 ∈ {∅} (𝑤𝑤) ⊆ ∅)))
5747, 50, 563anbi123d 1437 . . . . . 6 (𝑣 = ∅ → ((∀𝑤 ∈ 𝒫 (∅ × ∅)(𝑣𝑤𝑤 ∈ {∅}) ∧ ∀𝑤 ∈ {∅} (𝑣𝑤) ∈ {∅} ∧ (( I ↾ ∅) ⊆ 𝑣𝑣 ∈ {∅} ∧ ∃𝑤 ∈ {∅} (𝑤𝑤) ⊆ 𝑣)) ↔ (∀𝑤 ∈ 𝒫 (∅ × ∅)(∅ ⊆ 𝑤𝑤 ∈ {∅}) ∧ ∀𝑤 ∈ {∅} (∅ ∩ 𝑤) ∈ {∅} ∧ (( I ↾ ∅) ⊆ ∅ ∧ ∅ ∈ {∅} ∧ ∃𝑤 ∈ {∅} (𝑤𝑤) ⊆ ∅))))
581, 57ralsn 4569 . . . . 5 (∀𝑣 ∈ {∅} (∀𝑤 ∈ 𝒫 (∅ × ∅)(𝑣𝑤𝑤 ∈ {∅}) ∧ ∀𝑤 ∈ {∅} (𝑣𝑤) ∈ {∅} ∧ (( I ↾ ∅) ⊆ 𝑣𝑣 ∈ {∅} ∧ ∃𝑤 ∈ {∅} (𝑤𝑤) ⊆ 𝑣)) ↔ (∀𝑤 ∈ 𝒫 (∅ × ∅)(∅ ⊆ 𝑤𝑤 ∈ {∅}) ∧ ∀𝑤 ∈ {∅} (∅ ∩ 𝑤) ∈ {∅} ∧ (( I ↾ ∅) ⊆ ∅ ∧ ∅ ∈ {∅} ∧ ∃𝑤 ∈ {∅} (𝑤𝑤) ⊆ ∅)))
5927, 33, 44, 58mpbir3an 1342 . . . 4 𝑣 ∈ {∅} (∀𝑤 ∈ 𝒫 (∅ × ∅)(𝑣𝑤𝑤 ∈ {∅}) ∧ ∀𝑤 ∈ {∅} (𝑣𝑤) ∈ {∅} ∧ (( I ↾ ∅) ⊆ 𝑣𝑣 ∈ {∅} ∧ ∃𝑤 ∈ {∅} (𝑤𝑤) ⊆ 𝑣))
60 isust 22948 . . . . 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 1342 . . 3 {∅} ∈ (UnifOn‘∅)
63 snssi 4693 . . 3 ({∅} ∈ (UnifOn‘∅) → {{∅}} ⊆ (UnifOn‘∅))
6462, 63ax-mp 5 . 2 {{∅}} ⊆ (UnifOn‘∅)
6516, 64eqssi 3891 1 (UnifOn‘∅) = {{∅}}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  w3a 1088   = wceq 1542  wcel 2113  wral 3053  wrex 3054  Vcvv 3397  cin 3840  wss 3841  c0 4209  𝒫 cpw 4485  {csn 4513   I cid 5424   × cxp 5517  ccnv 5518  cres 5521  ccom 5523  cfv 6333  UnifOncust 22944
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3399  df-sbc 3680  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-br 5028  df-opab 5090  df-mpt 5108  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-res 5531  df-iota 6291  df-fun 6335  df-fv 6341  df-ust 22945
This theorem is referenced by:  isusp  23006
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