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Theorem exmidn0m 4185
Description: Excluded middle is equivalent to any set being empty or inhabited. (Contributed by Jim Kingdon, 5-Mar-2023.)
Assertion
Ref Expression
exmidn0m (EXMID ↔ ∀𝑥(𝑥 = ∅ ∨ ∃𝑦 𝑦𝑥))
Distinct variable group:   𝑥,𝑦

Proof of Theorem exmidn0m
StepHypRef Expression
1 simpr 109 . . . . 5 ((EXMID ∧ ∃𝑦 𝑦𝑥) → ∃𝑦 𝑦𝑥)
21olcd 729 . . . 4 ((EXMID ∧ ∃𝑦 𝑦𝑥) → (𝑥 = ∅ ∨ ∃𝑦 𝑦𝑥))
3 notm0 3434 . . . . . . 7 (¬ ∃𝑦 𝑦𝑥𝑥 = ∅)
43biimpi 119 . . . . . 6 (¬ ∃𝑦 𝑦𝑥𝑥 = ∅)
54adantl 275 . . . . 5 ((EXMID ∧ ¬ ∃𝑦 𝑦𝑥) → 𝑥 = ∅)
65orcd 728 . . . 4 ((EXMID ∧ ¬ ∃𝑦 𝑦𝑥) → (𝑥 = ∅ ∨ ∃𝑦 𝑦𝑥))
7 exmidexmid 4180 . . . . 5 (EXMIDDECID𝑦 𝑦𝑥)
8 exmiddc 831 . . . . 5 (DECID𝑦 𝑦𝑥 → (∃𝑦 𝑦𝑥 ∨ ¬ ∃𝑦 𝑦𝑥))
97, 8syl 14 . . . 4 (EXMID → (∃𝑦 𝑦𝑥 ∨ ¬ ∃𝑦 𝑦𝑥))
102, 6, 9mpjaodan 793 . . 3 (EXMID → (𝑥 = ∅ ∨ ∃𝑦 𝑦𝑥))
1110alrimiv 1867 . 2 (EXMID → ∀𝑥(𝑥 = ∅ ∨ ∃𝑦 𝑦𝑥))
12 orc 707 . . . . . 6 (𝑥 = ∅ → (𝑥 = ∅ ∨ 𝑥 = {∅}))
1312a1d 22 . . . . 5 (𝑥 = ∅ → (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
14 sssnm 3739 . . . . . . . 8 (∃𝑦 𝑦𝑥 → (𝑥 ⊆ {∅} ↔ 𝑥 = {∅}))
1514biimpa 294 . . . . . . 7 ((∃𝑦 𝑦𝑥𝑥 ⊆ {∅}) → 𝑥 = {∅})
1615olcd 729 . . . . . 6 ((∃𝑦 𝑦𝑥𝑥 ⊆ {∅}) → (𝑥 = ∅ ∨ 𝑥 = {∅}))
1716ex 114 . . . . 5 (∃𝑦 𝑦𝑥 → (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
1813, 17jaoi 711 . . . 4 ((𝑥 = ∅ ∨ ∃𝑦 𝑦𝑥) → (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
1918alimi 1448 . . 3 (∀𝑥(𝑥 = ∅ ∨ ∃𝑦 𝑦𝑥) → ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
20 exmid01 4182 . . 3 (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
2119, 20sylibr 133 . 2 (∀𝑥(𝑥 = ∅ ∨ ∃𝑦 𝑦𝑥) → EXMID)
2211, 21impbii 125 1 (EXMID ↔ ∀𝑥(𝑥 = ∅ ∨ ∃𝑦 𝑦𝑥))
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  wex 1485  wss 3121  c0 3414  {csn 3581  EXMIDwem 4178
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-14 2144  ax-ext 2152  ax-sep 4105  ax-nul 4113  ax-pow 4158
This theorem depends on definitions:  df-bi 116  df-dc 830  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rab 2457  df-v 2732  df-dif 3123  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3566  df-sn 3587  df-exmid 4179
This theorem is referenced by:  exmidsssn  4186
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