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Theorem exmid01 4159
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 variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-exmid 4156 . 2 (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥))
2 df-dc 821 . . . . 5 (DECID ∅ ∈ 𝑥 ↔ (∅ ∈ 𝑥 ∨ ¬ ∅ ∈ 𝑥))
3 orcom 718 . . . . . 6 ((∅ ∈ 𝑥 ∨ ¬ ∅ ∈ 𝑥) ↔ (¬ ∅ ∈ 𝑥 ∨ ∅ ∈ 𝑥))
4 simpll 519 . . . . . . . . . . . . . 14 (((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) ∧ 𝑦𝑥) → 𝑥 ⊆ {∅})
5 simpr 109 . . . . . . . . . . . . . 14 (((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) ∧ 𝑦𝑥) → 𝑦𝑥)
64, 5sseldd 3129 . . . . . . . . . . . . 13 (((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) ∧ 𝑦𝑥) → 𝑦 ∈ {∅})
7 velsn 3577 . . . . . . . . . . . . 13 (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
86, 7sylib 121 . . . . . . . . . . . 12 (((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) ∧ 𝑦𝑥) → 𝑦 = ∅)
98, 5eqeltrrd 2235 . . . . . . . . . . 11 (((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) ∧ 𝑦𝑥) → ∅ ∈ 𝑥)
10 simplr 520 . . . . . . . . . . 11 (((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) ∧ 𝑦𝑥) → ¬ ∅ ∈ 𝑥)
119, 10pm2.65da 651 . . . . . . . . . 10 ((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) → ¬ 𝑦𝑥)
1211eq0rdv 3438 . . . . . . . . 9 ((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) → 𝑥 = ∅)
1312ex 114 . . . . . . . 8 (𝑥 ⊆ {∅} → (¬ ∅ ∈ 𝑥𝑥 = ∅))
14 noel 3398 . . . . . . . . 9 ¬ ∅ ∈ ∅
15 eleq2 2221 . . . . . . . . 9 (𝑥 = ∅ → (∅ ∈ 𝑥 ↔ ∅ ∈ ∅))
1614, 15mtbiri 665 . . . . . . . 8 (𝑥 = ∅ → ¬ ∅ ∈ 𝑥)
1713, 16impbid1 141 . . . . . . 7 (𝑥 ⊆ {∅} → (¬ ∅ ∈ 𝑥𝑥 = ∅))
18 ss1o0el1 4158 . . . . . . 7 (𝑥 ⊆ {∅} → (∅ ∈ 𝑥𝑥 = {∅}))
1917, 18orbi12d 783 . . . . . 6 (𝑥 ⊆ {∅} → ((¬ ∅ ∈ 𝑥 ∨ ∅ ∈ 𝑥) ↔ (𝑥 = ∅ ∨ 𝑥 = {∅})))
203, 19syl5bb 191 . . . . 5 (𝑥 ⊆ {∅} → ((∅ ∈ 𝑥 ∨ ¬ ∅ ∈ 𝑥) ↔ (𝑥 = ∅ ∨ 𝑥 = {∅})))
212, 20syl5bb 191 . . . 4 (𝑥 ⊆ {∅} → (DECID ∅ ∈ 𝑥 ↔ (𝑥 = ∅ ∨ 𝑥 = {∅})))
2221pm5.74i 179 . . 3 ((𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥) ↔ (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
2322albii 1450 . 2 (∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥) ↔ ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
241, 23bitri 183 1 (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 698  DECID wdc 820  wal 1333   = wceq 1335  wcel 2128  wss 3102  c0 3394  {csn 3560  EXMIDwem 4155
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139  ax-nul 4090
This theorem depends on definitions:  df-bi 116  df-dc 821  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-dif 3104  df-in 3108  df-ss 3115  df-nul 3395  df-sn 3566  df-exmid 4156
This theorem is referenced by:  exmid1dc  4161  exmidn0m  4162  exmidsssn  4163  exmidpw  6850  exmidpweq  6851  exmidomni  7080  ss1oel2o  13552  exmidsbthrlem  13580  sbthom  13584
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