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Theorem exmidn0m 4222
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 735 . . . 4 ((EXMID ∧ ∃𝑦 𝑦𝑥) → (𝑥 = ∅ ∨ ∃𝑦 𝑦𝑥))
3 notm0 3458 . . . . . . 7 (¬ ∃𝑦 𝑦𝑥𝑥 = ∅)
43biimpi 120 . . . . . 6 (¬ ∃𝑦 𝑦𝑥𝑥 = ∅)
54adantl 277 . . . . 5 ((EXMID ∧ ¬ ∃𝑦 𝑦𝑥) → 𝑥 = ∅)
65orcd 734 . . . 4 ((EXMID ∧ ¬ ∃𝑦 𝑦𝑥) → (𝑥 = ∅ ∨ ∃𝑦 𝑦𝑥))
7 exmidexmid 4217 . . . . 5 (EXMIDDECID𝑦 𝑦𝑥)
8 exmiddc 837 . . . . 5 (DECID𝑦 𝑦𝑥 → (∃𝑦 𝑦𝑥 ∨ ¬ ∃𝑦 𝑦𝑥))
97, 8syl 14 . . . 4 (EXMID → (∃𝑦 𝑦𝑥 ∨ ¬ ∃𝑦 𝑦𝑥))
102, 6, 9mpjaodan 799 . . 3 (EXMID → (𝑥 = ∅ ∨ ∃𝑦 𝑦𝑥))
1110alrimiv 1885 . 2 (EXMID → ∀𝑥(𝑥 = ∅ ∨ ∃𝑦 𝑦𝑥))
12 orc 713 . . . . . 6 (𝑥 = ∅ → (𝑥 = ∅ ∨ 𝑥 = {∅}))
1312a1d 22 . . . . 5 (𝑥 = ∅ → (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
14 sssnm 3772 . . . . . . . 8 (∃𝑦 𝑦𝑥 → (𝑥 ⊆ {∅} ↔ 𝑥 = {∅}))
1514biimpa 296 . . . . . . 7 ((∃𝑦 𝑦𝑥𝑥 ⊆ {∅}) → 𝑥 = {∅})
1615olcd 735 . . . . . 6 ((∃𝑦 𝑦𝑥𝑥 ⊆ {∅}) → (𝑥 = ∅ ∨ 𝑥 = {∅}))
1716ex 115 . . . . 5 (∃𝑦 𝑦𝑥 → (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
1813, 17jaoi 717 . . . 4 ((𝑥 = ∅ ∨ ∃𝑦 𝑦𝑥) → (𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
1918alimi 1466 . . 3 (∀𝑥(𝑥 = ∅ ∨ ∃𝑦 𝑦𝑥) → ∀𝑥(𝑥 ⊆ {∅} → (𝑥 = ∅ ∨ 𝑥 = {∅})))
20 exmid01 4219 . . 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 709  DECID wdc 835  wal 1362   = wceq 1364  wex 1503  wss 3144  c0 3437  {csn 3610  EXMIDwem 4215
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-nul 4147  ax-pow 4195
This theorem depends on definitions:  df-bi 117  df-dc 836  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rab 2477  df-v 2754  df-dif 3146  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3595  df-sn 3616  df-exmid 4216
This theorem is referenced by:  exmidsssn  4223
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