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Theorem exmid0el 4095
 Description: Excluded middle is equivalent to decidability of ∅ being an element of an arbitrary set. (Contributed by Jim Kingdon, 18-Jun-2022.)
Assertion
Ref Expression
exmid0el (EXMID ↔ ∀𝑥DECID ∅ ∈ 𝑥)

Proof of Theorem exmid0el
StepHypRef Expression
1 exmidexmid 4088 . . 3 (EXMIDDECID ∅ ∈ 𝑥)
21alrimiv 1828 . 2 (EXMID → ∀𝑥DECID ∅ ∈ 𝑥)
3 ax-1 6 . . . 4 (DECID ∅ ∈ 𝑥 → (𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥))
43alimi 1414 . . 3 (∀𝑥DECID ∅ ∈ 𝑥 → ∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥))
5 df-exmid 4087 . . 3 (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥))
64, 5sylibr 133 . 2 (∀𝑥DECID ∅ ∈ 𝑥EXMID)
72, 6impbii 125 1 (EXMID ↔ ∀𝑥DECID ∅ ∈ 𝑥)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104  DECID wdc 802  ∀wal 1312   ∈ wcel 1463   ⊆ wss 3039  ∅c0 3331  {csn 3495  EXMIDwem 4086 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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-nul 4022  ax-pow 4066 This theorem depends on definitions:  df-bi 116  df-dc 803  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-rab 2400  df-v 2660  df-dif 3041  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-exmid 4087 This theorem is referenced by:  exmidel  4096
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