Theorem exmid0el 4294

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 4286	. . 3 ⊢ (EXMID → DECID ∅ ∈ 𝑥)
2	1	alrimiv 1922	. 2 ⊢ (EXMID → ∀𝑥DECID ∅ ∈ 𝑥)
3		ax-1 6	. . . 4 ⊢ (DECID ∅ ∈ 𝑥 → (𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥))
4	3	alimi 1503	. . 3 ⊢ (∀𝑥DECID ∅ ∈ 𝑥 → ∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥))
5		df-exmid 4285	. . 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 841  ∀wal 1395   ∈ wcel 2202   ⊆ wss 3200  ∅c0 3494  {csn 3669  EXMIDwem 4284

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264

This theorem depends on definitions:  df-bi 117  df-dc 842  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rab 2519  df-v 2804  df-dif 3202  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-exmid 4285

This theorem is referenced by:  exmidel  4295