Theorem exmid0el 4206

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 4198	. . 3 ⊢ (EXMID → DECID ∅ ∈ 𝑥)
2	1	alrimiv 1874	. 2 ⊢ (EXMID → ∀𝑥DECID ∅ ∈ 𝑥)
3		ax-1 6	. . . 4 ⊢ (DECID ∅ ∈ 𝑥 → (𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥))
4	3	alimi 1455	. . 3 ⊢ (∀𝑥DECID ∅ ∈ 𝑥 → ∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥))
5		df-exmid 4197	. . 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 834  ∀wal 1351   ∈ wcel 2148   ⊆ wss 3131  ∅c0 3424  {csn 3594  EXMIDwem 4196

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-nul 4131  ax-pow 4176

This theorem depends on definitions:  df-bi 117  df-dc 835  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rab 2464  df-v 2741  df-dif 3133  df-in 3137  df-ss 3144  df-nul 3425  df-pw 3579  df-sn 3600  df-exmid 4197

This theorem is referenced by:  exmidel  4207