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Theorem exmid01 3995
Description: Excluded middle is equivalent to saying any subset of {∅} is either or {∅}. (Contributed by BJ and Jim Kingdon, 18-Jun-2022.)
Assertion
Ref Expression
exmid01 (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))

Proof of Theorem exmid01
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-exmid 3993 . 2 (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥))
2 df-dc 777 . . . . 5 (DECID ∅ ∈ 𝑥 ↔ (∅ ∈ 𝑥 ∨ ¬ ∅ ∈ 𝑥))
3 orcom 680 . . . . . 6 ((∅ ∈ 𝑥 ∨ ¬ ∅ ∈ 𝑥) ↔ (¬ ∅ ∈ 𝑥 ∨ ∅ ∈ 𝑥))
4 simpll 496 . . . . . . . . . . . . . 14 (((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) ∧ 𝑦𝑥) → 𝑥 ⊆ {∅})
5 simpr 108 . . . . . . . . . . . . . 14 (((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) ∧ 𝑦𝑥) → 𝑦𝑥)
64, 5sseldd 3011 . . . . . . . . . . . . 13 (((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) ∧ 𝑦𝑥) → 𝑦 ∈ {∅})
7 velsn 3439 . . . . . . . . . . . . 13 (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
86, 7sylib 120 . . . . . . . . . . . 12 (((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) ∧ 𝑦𝑥) → 𝑦 = ∅)
98, 5eqeltrrd 2160 . . . . . . . . . . 11 (((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) ∧ 𝑦𝑥) → ∅ ∈ 𝑥)
10 simplr 497 . . . . . . . . . . 11 (((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) ∧ 𝑦𝑥) → ¬ ∅ ∈ 𝑥)
119, 10pm2.65da 620 . . . . . . . . . 10 ((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) → ¬ 𝑦𝑥)
1211eq0rdv 3309 . . . . . . . . 9 ((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) → 𝑥 = ∅)
1312ex 113 . . . . . . . 8 (𝑥 ⊆ {∅} → (¬ ∅ ∈ 𝑥𝑥 = ∅))
14 noel 3273 . . . . . . . . 9 ¬ ∅ ∈ ∅
15 eleq2 2146 . . . . . . . . 9 (𝑥 = ∅ → (∅ ∈ 𝑥 ↔ ∅ ∈ ∅))
1614, 15mtbiri 633 . . . . . . . 8 (𝑥 = ∅ → ¬ ∅ ∈ 𝑥)
1713, 16impbid1 140 . . . . . . 7 (𝑥 ⊆ {∅} → (¬ ∅ ∈ 𝑥𝑥 = ∅))
18 elex2 2626 . . . . . . . . . 10 (∅ ∈ 𝑥 → ∃𝑧 𝑧𝑥)
19 sssnm 3572 . . . . . . . . . 10 (∃𝑧 𝑧𝑥 → (𝑥 ⊆ {∅} ↔ 𝑥 = {∅}))
2018, 19syl 14 . . . . . . . . 9 (∅ ∈ 𝑥 → (𝑥 ⊆ {∅} ↔ 𝑥 = {∅}))
2120biimpcd 157 . . . . . . . 8 (𝑥 ⊆ {∅} → (∅ ∈ 𝑥𝑥 = {∅}))
22 0ex 3931 . . . . . . . . . 10 ∅ ∈ V
2322snid 3449 . . . . . . . . 9 ∅ ∈ {∅}
24 eleq2 2146 . . . . . . . . 9 (𝑥 = {∅} → (∅ ∈ 𝑥 ↔ ∅ ∈ {∅}))
2523, 24mpbiri 166 . . . . . . . 8 (𝑥 = {∅} → ∅ ∈ 𝑥)
2621, 25impbid1 140 . . . . . . 7 (𝑥 ⊆ {∅} → (∅ ∈ 𝑥𝑥 = {∅}))
2717, 26orbi12d 740 . . . . . 6 (𝑥 ⊆ {∅} → ((¬ ∅ ∈ 𝑥 ∨ ∅ ∈ 𝑥) ↔ (𝑥 = ∅ ∨ 𝑥 = {∅})))
283, 27syl5bb 190 . . . . 5 (𝑥 ⊆ {∅} → ((∅ ∈ 𝑥 ∨ ¬ ∅ ∈ 𝑥) ↔ (𝑥 = ∅ ∨ 𝑥 = {∅})))
292, 28syl5bb 190 . . . 4 (𝑥 ⊆ {∅} → (DECID ∅ ∈ 𝑥 ↔ (𝑥 = ∅ ∨ 𝑥 = {∅})))
3029pm5.74i 178 . . 3 ((𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥) ↔ (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
3130albii 1400 . 2 (∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥) ↔ ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
321, 31bitri 182 1 (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wo 662  DECID wdc 776  wal 1283   = wceq 1285  wex 1422  wcel 1434  wss 2984  c0 3269  {csn 3422  EXMIDwem 3992
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-nul 3930
This theorem depends on definitions:  df-bi 115  df-dc 777  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-dif 2986  df-in 2990  df-ss 2997  df-nul 3270  df-sn 3428  df-exmid 3993
This theorem is referenced by:  exmidpw  6549  exmidomni  6701
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