Theorem exmid0el 4287

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 4279	. . 3 ⊢ (EXMID → DECID ∅ ∈ 𝑥)
2	1	alrimiv 1920	. 2 ⊢ (EXMID → ∀𝑥DECID ∅ ∈ 𝑥)
3		ax-1 6	. . . 4 ⊢ (DECID ∅ ∈ 𝑥 → (𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥))
4	3	alimi 1501	. . 3 ⊢ (∀𝑥DECID ∅ ∈ 𝑥 → ∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥))
5		df-exmid 4278	. . 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 839  ∀wal 1393   ∈ wcel 2200   ⊆ wss 3197  ∅c0 3491  {csn 3666  EXMIDwem 4277

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-nul 4209  ax-pow 4257

This theorem depends on definitions:  df-bi 117  df-dc 840  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rab 2517  df-v 2801  df-dif 3199  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-exmid 4278

This theorem is referenced by:  exmidel  4288