ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  exmidsssnc GIF version

Theorem exmidsssnc 4200
Description: Excluded middle in terms of subsets of a singleton. This is similar to exmid01 4195 but lets you choose any set as the element of the singleton rather than just . It is similar to exmidsssn 4199 but for a particular set 𝐵 rather than all sets. (Contributed by Jim Kingdon, 29-Jul-2023.)
Assertion
Ref Expression
exmidsssnc (𝐵𝑉 → (EXMID ↔ ∀𝑥(𝑥 ⊆ {𝐵} → (𝑥 = ∅ ∨ 𝑥 = {𝐵}))))
Distinct variable groups:   𝑥,𝐵   𝑥,𝑉

Proof of Theorem exmidsssnc
Dummy variables 𝑢 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 exmidsssn 4199 . . . 4 (EXMID ↔ ∀𝑥𝑢(𝑥 ⊆ {𝑢} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝑢})))
2 sneq 3602 . . . . . . . 8 (𝑢 = 𝐵 → {𝑢} = {𝐵})
32sseq2d 3185 . . . . . . 7 (𝑢 = 𝐵 → (𝑥 ⊆ {𝑢} ↔ 𝑥 ⊆ {𝐵}))
42eqeq2d 2189 . . . . . . . 8 (𝑢 = 𝐵 → (𝑥 = {𝑢} ↔ 𝑥 = {𝐵}))
54orbi2d 790 . . . . . . 7 (𝑢 = 𝐵 → ((𝑥 = ∅ ∨ 𝑥 = {𝑢}) ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐵})))
63, 5bibi12d 235 . . . . . 6 (𝑢 = 𝐵 → ((𝑥 ⊆ {𝑢} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝑢})) ↔ (𝑥 ⊆ {𝐵} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐵}))))
76spcgv 2824 . . . . 5 (𝐵𝑉 → (∀𝑢(𝑥 ⊆ {𝑢} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝑢})) → (𝑥 ⊆ {𝐵} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐵}))))
87alimdv 1879 . . . 4 (𝐵𝑉 → (∀𝑥𝑢(𝑥 ⊆ {𝑢} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝑢})) → ∀𝑥(𝑥 ⊆ {𝐵} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐵}))))
91, 8biimtrid 152 . . 3 (𝐵𝑉 → (EXMID → ∀𝑥(𝑥 ⊆ {𝐵} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐵}))))
10 biimp 118 . . . 4 ((𝑥 ⊆ {𝐵} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐵})) → (𝑥 ⊆ {𝐵} → (𝑥 = ∅ ∨ 𝑥 = {𝐵})))
1110alimi 1455 . . 3 (∀𝑥(𝑥 ⊆ {𝐵} ↔ (𝑥 = ∅ ∨ 𝑥 = {𝐵})) → ∀𝑥(𝑥 ⊆ {𝐵} → (𝑥 = ∅ ∨ 𝑥 = {𝐵})))
129, 11syl6 33 . 2 (𝐵𝑉 → (EXMID → ∀𝑥(𝑥 ⊆ {𝐵} → (𝑥 = ∅ ∨ 𝑥 = {𝐵}))))
13 ssrab2 3240 . . . . . . . . 9 {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} ⊆ {𝐵}
14 snexg 4181 . . . . . . . . . 10 (𝐵𝑉 → {𝐵} ∈ V)
15 rabexg 4143 . . . . . . . . . 10 ({𝐵} ∈ V → {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} ∈ V)
16 sseq1 3178 . . . . . . . . . . . 12 (𝑥 = {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} → (𝑥 ⊆ {𝐵} ↔ {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} ⊆ {𝐵}))
17 eqeq1 2184 . . . . . . . . . . . . 13 (𝑥 = {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} → (𝑥 = ∅ ↔ {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = ∅))
18 eqeq1 2184 . . . . . . . . . . . . 13 (𝑥 = {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} → (𝑥 = {𝐵} ↔ {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = {𝐵}))
1917, 18orbi12d 793 . . . . . . . . . . . 12 (𝑥 = {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} → ((𝑥 = ∅ ∨ 𝑥 = {𝐵}) ↔ ({𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = ∅ ∨ {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = {𝐵})))
2016, 19imbi12d 234 . . . . . . . . . . 11 (𝑥 = {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} → ((𝑥 ⊆ {𝐵} → (𝑥 = ∅ ∨ 𝑥 = {𝐵})) ↔ ({𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} ⊆ {𝐵} → ({𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = ∅ ∨ {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = {𝐵}))))
2120spcgv 2824 . . . . . . . . . 10 ({𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} ∈ V → (∀𝑥(𝑥 ⊆ {𝐵} → (𝑥 = ∅ ∨ 𝑥 = {𝐵})) → ({𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} ⊆ {𝐵} → ({𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = ∅ ∨ {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = {𝐵}))))
2214, 15, 213syl 17 . . . . . . . . 9 (𝐵𝑉 → (∀𝑥(𝑥 ⊆ {𝐵} → (𝑥 = ∅ ∨ 𝑥 = {𝐵})) → ({𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} ⊆ {𝐵} → ({𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = ∅ ∨ {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = {𝐵}))))
2313, 22mpii 44 . . . . . . . 8 (𝐵𝑉 → (∀𝑥(𝑥 ⊆ {𝐵} → (𝑥 = ∅ ∨ 𝑥 = {𝐵})) → ({𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = ∅ ∨ {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = {𝐵})))
24 rabeq0 3452 . . . . . . . . . . 11 ({𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = ∅ ↔ ∀𝑧 ∈ {𝐵} ¬ ∅ ∈ 𝑦)
25 snmg 3709 . . . . . . . . . . . 12 (𝐵𝑉 → ∃𝑤 𝑤 ∈ {𝐵})
26 r19.3rmv 3513 . . . . . . . . . . . 12 (∃𝑤 𝑤 ∈ {𝐵} → (¬ ∅ ∈ 𝑦 ↔ ∀𝑧 ∈ {𝐵} ¬ ∅ ∈ 𝑦))
2725, 26syl 14 . . . . . . . . . . 11 (𝐵𝑉 → (¬ ∅ ∈ 𝑦 ↔ ∀𝑧 ∈ {𝐵} ¬ ∅ ∈ 𝑦))
2824, 27bitr4id 199 . . . . . . . . . 10 (𝐵𝑉 → ({𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = ∅ ↔ ¬ ∅ ∈ 𝑦))
2928biimpd 144 . . . . . . . . 9 (𝐵𝑉 → ({𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = ∅ → ¬ ∅ ∈ 𝑦))
30 snidg 3620 . . . . . . . . . . . . 13 (𝐵𝑉𝐵 ∈ {𝐵})
3130adantr 276 . . . . . . . . . . . 12 ((𝐵𝑉 ∧ {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = {𝐵}) → 𝐵 ∈ {𝐵})
32 simpr 110 . . . . . . . . . . . 12 ((𝐵𝑉 ∧ {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = {𝐵}) → {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = {𝐵})
3331, 32eleqtrrd 2257 . . . . . . . . . . 11 ((𝐵𝑉 ∧ {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = {𝐵}) → 𝐵 ∈ {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦})
34 biidd 172 . . . . . . . . . . . . 13 (𝑧 = 𝐵 → (∅ ∈ 𝑦 ↔ ∅ ∈ 𝑦))
3534elrab 2893 . . . . . . . . . . . 12 (𝐵 ∈ {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} ↔ (𝐵 ∈ {𝐵} ∧ ∅ ∈ 𝑦))
3635simprbi 275 . . . . . . . . . . 11 (𝐵 ∈ {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} → ∅ ∈ 𝑦)
3733, 36syl 14 . . . . . . . . . 10 ((𝐵𝑉 ∧ {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = {𝐵}) → ∅ ∈ 𝑦)
3837ex 115 . . . . . . . . 9 (𝐵𝑉 → ({𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = {𝐵} → ∅ ∈ 𝑦))
3929, 38orim12d 786 . . . . . . . 8 (𝐵𝑉 → (({𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = ∅ ∨ {𝑧 ∈ {𝐵} ∣ ∅ ∈ 𝑦} = {𝐵}) → (¬ ∅ ∈ 𝑦 ∨ ∅ ∈ 𝑦)))
4023, 39syld 45 . . . . . . 7 (𝐵𝑉 → (∀𝑥(𝑥 ⊆ {𝐵} → (𝑥 = ∅ ∨ 𝑥 = {𝐵})) → (¬ ∅ ∈ 𝑦 ∨ ∅ ∈ 𝑦)))
41 orcom 728 . . . . . . 7 ((¬ ∅ ∈ 𝑦 ∨ ∅ ∈ 𝑦) ↔ (∅ ∈ 𝑦 ∨ ¬ ∅ ∈ 𝑦))
4240, 41syl6ib 161 . . . . . 6 (𝐵𝑉 → (∀𝑥(𝑥 ⊆ {𝐵} → (𝑥 = ∅ ∨ 𝑥 = {𝐵})) → (∅ ∈ 𝑦 ∨ ¬ ∅ ∈ 𝑦)))
43 df-dc 835 . . . . . 6 (DECID ∅ ∈ 𝑦 ↔ (∅ ∈ 𝑦 ∨ ¬ ∅ ∈ 𝑦))
4442, 43syl6ibr 162 . . . . 5 (𝐵𝑉 → (∀𝑥(𝑥 ⊆ {𝐵} → (𝑥 = ∅ ∨ 𝑥 = {𝐵})) → DECID ∅ ∈ 𝑦))
4544a1dd 48 . . . 4 (𝐵𝑉 → (∀𝑥(𝑥 ⊆ {𝐵} → (𝑥 = ∅ ∨ 𝑥 = {𝐵})) → (𝑦 ⊆ {∅} → DECID ∅ ∈ 𝑦)))
4645alrimdv 1876 . . 3 (𝐵𝑉 → (∀𝑥(𝑥 ⊆ {𝐵} → (𝑥 = ∅ ∨ 𝑥 = {𝐵})) → ∀𝑦(𝑦 ⊆ {∅} → DECID ∅ ∈ 𝑦)))
47 df-exmid 4192 . . 3 (EXMID ↔ ∀𝑦(𝑦 ⊆ {∅} → DECID ∅ ∈ 𝑦))
4846, 47syl6ibr 162 . 2 (𝐵𝑉 → (∀𝑥(𝑥 ⊆ {𝐵} → (𝑥 = ∅ ∨ 𝑥 = {𝐵})) → EXMID))
4912, 48impbid 129 1 (𝐵𝑉 → (EXMID ↔ ∀𝑥(𝑥 ⊆ {𝐵} → (𝑥 = ∅ ∨ 𝑥 = {𝐵}))))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 708  DECID wdc 834  wal 1351   = wceq 1353  wex 1492  wcel 2148  wral 2455  {crab 2459  Vcvv 2737  wss 3129  c0 3422  {csn 3591  EXMIDwem 4191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-nul 4126  ax-pow 4171
This theorem depends on definitions:  df-bi 117  df-dc 835  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rab 2464  df-v 2739  df-dif 3131  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-exmid 4192
This theorem is referenced by:  exmidunben  12410
  Copyright terms: Public domain W3C validator