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Theorem exmidn0m 4203
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 110 . . . . 5 ((EXMID ∧ ∃𝑦 𝑦𝑥) → ∃𝑦 𝑦𝑥)
21olcd 734 . . . 4 ((EXMID ∧ ∃𝑦 𝑦𝑥) → (𝑥 = ∅ ∨ ∃𝑦 𝑦𝑥))
3 notm0 3445 . . . . . . 7 (¬ ∃𝑦 𝑦𝑥𝑥 = ∅)
43biimpi 120 . . . . . 6 (¬ ∃𝑦 𝑦𝑥𝑥 = ∅)
54adantl 277 . . . . 5 ((EXMID ∧ ¬ ∃𝑦 𝑦𝑥) → 𝑥 = ∅)
65orcd 733 . . . 4 ((EXMID ∧ ¬ ∃𝑦 𝑦𝑥) → (𝑥 = ∅ ∨ ∃𝑦 𝑦𝑥))
7 exmidexmid 4198 . . . . 5 (EXMIDDECID𝑦 𝑦𝑥)
8 exmiddc 836 . . . . 5 (DECID𝑦 𝑦𝑥 → (∃𝑦 𝑦𝑥 ∨ ¬ ∃𝑦 𝑦𝑥))
97, 8syl 14 . . . 4 (EXMID → (∃𝑦 𝑦𝑥 ∨ ¬ ∃𝑦 𝑦𝑥))
102, 6, 9mpjaodan 798 . . 3 (EXMID → (𝑥 = ∅ ∨ ∃𝑦 𝑦𝑥))
1110alrimiv 1874 . 2 (EXMID → ∀𝑥(𝑥 = ∅ ∨ ∃𝑦 𝑦𝑥))
12 orc 712 . . . . . 6 (𝑥 = ∅ → (𝑥 = ∅ ∨ 𝑥 = {∅}))
1312a1d 22 . . . . 5 (𝑥 = ∅ → (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
14 sssnm 3756 . . . . . . . 8 (∃𝑦 𝑦𝑥 → (𝑥 ⊆ {∅} ↔ 𝑥 = {∅}))
1514biimpa 296 . . . . . . 7 ((∃𝑦 𝑦𝑥𝑥 ⊆ {∅}) → 𝑥 = {∅})
1615olcd 734 . . . . . 6 ((∃𝑦 𝑦𝑥𝑥 ⊆ {∅}) → (𝑥 = ∅ ∨ 𝑥 = {∅}))
1716ex 115 . . . . 5 (∃𝑦 𝑦𝑥 → (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
1813, 17jaoi 716 . . . 4 ((𝑥 = ∅ ∨ ∃𝑦 𝑦𝑥) → (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
1918alimi 1455 . . 3 (∀𝑥(𝑥 = ∅ ∨ ∃𝑦 𝑦𝑥) → ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
20 exmid01 4200 . . 3 (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
2119, 20sylibr 134 . 2 (∀𝑥(𝑥 = ∅ ∨ ∃𝑦 𝑦𝑥) → EXMID)
2211, 21impbii 126 1 (EXMID ↔ ∀𝑥(𝑥 = ∅ ∨ ∃𝑦 𝑦𝑥))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 708  DECID wdc 834  wal 1351   = wceq 1353  wex 1492  wss 3131  c0 3424  {csn 3594  EXMIDwem 4196
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-nul 4131  ax-pow 4176
This theorem depends on definitions:  df-bi 117  df-dc 835  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rab 2464  df-v 2741  df-dif 3133  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-exmid 4197
This theorem is referenced by:  exmidsssn  4204
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