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Theorem exmid01 4184
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 4181 . 2 (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥))
2 df-dc 830 . . . . 5 (DECID ∅ ∈ 𝑥 ↔ (∅ ∈ 𝑥 ∨ ¬ ∅ ∈ 𝑥))
3 orcom 723 . . . . . 6 ((∅ ∈ 𝑥 ∨ ¬ ∅ ∈ 𝑥) ↔ (¬ ∅ ∈ 𝑥 ∨ ∅ ∈ 𝑥))
4 simpll 524 . . . . . . . . . . . . . 14 (((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) ∧ 𝑦𝑥) → 𝑥 ⊆ {∅})
5 simpr 109 . . . . . . . . . . . . . 14 (((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) ∧ 𝑦𝑥) → 𝑦𝑥)
64, 5sseldd 3148 . . . . . . . . . . . . 13 (((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) ∧ 𝑦𝑥) → 𝑦 ∈ {∅})
7 velsn 3600 . . . . . . . . . . . . 13 (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
86, 7sylib 121 . . . . . . . . . . . 12 (((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) ∧ 𝑦𝑥) → 𝑦 = ∅)
98, 5eqeltrrd 2248 . . . . . . . . . . 11 (((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) ∧ 𝑦𝑥) → ∅ ∈ 𝑥)
10 simplr 525 . . . . . . . . . . 11 (((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) ∧ 𝑦𝑥) → ¬ ∅ ∈ 𝑥)
119, 10pm2.65da 656 . . . . . . . . . 10 ((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) → ¬ 𝑦𝑥)
1211eq0rdv 3459 . . . . . . . . 9 ((𝑥 ⊆ {∅} ∧ ¬ ∅ ∈ 𝑥) → 𝑥 = ∅)
1312ex 114 . . . . . . . 8 (𝑥 ⊆ {∅} → (¬ ∅ ∈ 𝑥𝑥 = ∅))
14 noel 3418 . . . . . . . . 9 ¬ ∅ ∈ ∅
15 eleq2 2234 . . . . . . . . 9 (𝑥 = ∅ → (∅ ∈ 𝑥 ↔ ∅ ∈ ∅))
1614, 15mtbiri 670 . . . . . . . 8 (𝑥 = ∅ → ¬ ∅ ∈ 𝑥)
1713, 16impbid1 141 . . . . . . 7 (𝑥 ⊆ {∅} → (¬ ∅ ∈ 𝑥𝑥 = ∅))
18 ss1o0el1 4183 . . . . . . 7 (𝑥 ⊆ {∅} → (∅ ∈ 𝑥𝑥 = {∅}))
1917, 18orbi12d 788 . . . . . 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 1463 . 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 703  DECID wdc 829  wal 1346   = wceq 1348  wcel 2141  wss 3121  c0 3414  {csn 3583  EXMIDwem 4180
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-nul 4115
This theorem depends on definitions:  df-bi 116  df-dc 830  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134  df-nul 3415  df-sn 3589  df-exmid 4181
This theorem is referenced by:  exmid1dc  4186  exmidn0m  4187  exmidsssn  4188  exmidpw  6886  exmidpweq  6887  exmidomni  7118  ss1oel2o  14026  exmidsbthrlem  14054  sbthom  14058
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