Theorem exmid0el 4234

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

Step	Hyp	Ref	Expression
1		exmidexmid 4226	. . 3 ⊢ (EXMID → DECID ∅ ∈ 𝑥)
2	1	alrimiv 1885	. 2 ⊢ (EXMID → ∀𝑥DECID ∅ ∈ 𝑥)
3		ax-1 6	. . . 4 ⊢ (DECID ∅ ∈ 𝑥 → (𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥))
4	3	alimi 1466	. . 3 ⊢ (∀𝑥DECID ∅ ∈ 𝑥 → ∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥))
5		df-exmid 4225	. . 3 ⊢ (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥))
6	4, 5	sylibr 134	. 2 ⊢ (∀𝑥DECID ∅ ∈ 𝑥 → EXMID)
7	2, 6	impbii 126	1 ⊢ (EXMID ↔ ∀𝑥DECID ∅ ∈ 𝑥)

Syntax hints:   → wi 4   ↔ wb 105  DECID wdc 835  ∀wal 1362   ∈ wcel 2164   ⊆ wss 3154  ∅c0 3447  {csn 3619  EXMIDwem 4224

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 2167  ax-ext 2175  ax-sep 4148  ax-nul 4156  ax-pow 4204

This theorem depends on definitions:  df-bi 117  df-dc 836  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rab 2481  df-v 2762  df-dif 3156  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-exmid 4225

This theorem is referenced by:  exmidel  4235