Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  exmidn0m GIF version

Theorem exmidn0m 4124
 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 723 . . . 4 ((EXMID ∧ ∃𝑦 𝑦𝑥) → (𝑥 = ∅ ∨ ∃𝑦 𝑦𝑥))
3 notm0 3383 . . . . . . 7 (¬ ∃𝑦 𝑦𝑥𝑥 = ∅)
43biimpi 119 . . . . . 6 (¬ ∃𝑦 𝑦𝑥𝑥 = ∅)
54adantl 275 . . . . 5 ((EXMID ∧ ¬ ∃𝑦 𝑦𝑥) → 𝑥 = ∅)
65orcd 722 . . . 4 ((EXMID ∧ ¬ ∃𝑦 𝑦𝑥) → (𝑥 = ∅ ∨ ∃𝑦 𝑦𝑥))
7 exmidexmid 4120 . . . . 5 (EXMIDDECID𝑦 𝑦𝑥)
8 exmiddc 821 . . . . 5 (DECID𝑦 𝑦𝑥 → (∃𝑦 𝑦𝑥 ∨ ¬ ∃𝑦 𝑦𝑥))
97, 8syl 14 . . . 4 (EXMID → (∃𝑦 𝑦𝑥 ∨ ¬ ∃𝑦 𝑦𝑥))
102, 6, 9mpjaodan 787 . . 3 (EXMID → (𝑥 = ∅ ∨ ∃𝑦 𝑦𝑥))
1110alrimiv 1846 . 2 (EXMID → ∀𝑥(𝑥 = ∅ ∨ ∃𝑦 𝑦𝑥))
12 orc 701 . . . . . 6 (𝑥 = ∅ → (𝑥 = ∅ ∨ 𝑥 = {∅}))
1312a1d 22 . . . . 5 (𝑥 = ∅ → (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
14 sssnm 3681 . . . . . . . 8 (∃𝑦 𝑦𝑥 → (𝑥 ⊆ {∅} ↔ 𝑥 = {∅}))
1514biimpa 294 . . . . . . 7 ((∃𝑦 𝑦𝑥𝑥 ⊆ {∅}) → 𝑥 = {∅})
1615olcd 723 . . . . . 6 ((∃𝑦 𝑦𝑥𝑥 ⊆ {∅}) → (𝑥 = ∅ ∨ 𝑥 = {∅}))
1716ex 114 . . . . 5 (∃𝑦 𝑦𝑥 → (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
1813, 17jaoi 705 . . . 4 ((𝑥 = ∅ ∨ ∃𝑦 𝑦𝑥) → (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
1918alimi 1431 . . 3 (∀𝑥(𝑥 = ∅ ∨ ∃𝑦 𝑦𝑥) → ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
20 exmid01 4121 . . 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 697  DECID wdc 819  ∀wal 1329   = wceq 1331  ∃wex 1468   ⊆ wss 3071  ∅c0 3363  {csn 3527  EXMIDwem 4118 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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098 This theorem depends on definitions:  df-bi 116  df-dc 820  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rab 2425  df-v 2688  df-dif 3073  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-exmid 4119 This theorem is referenced by:  exmidsssn  4125
 Copyright terms: Public domain W3C validator