Theorem exmid0el 4237

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 4229	. . 3 ⊢ (EXMID → DECID ∅ ∈ 𝑥)
2	1	alrimiv 1888	. 2 ⊢ (EXMID → ∀𝑥DECID ∅ ∈ 𝑥)
3		ax-1 6	. . . 4 ⊢ (DECID ∅ ∈ 𝑥 → (𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥))
4	3	alimi 1469	. . 3 ⊢ (∀𝑥DECID ∅ ∈ 𝑥 → ∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥))
5		df-exmid 4228	. . 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 2167   ⊆ wss 3157  ∅c0 3450  {csn 3622  EXMIDwem 4227

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-nul 4159  ax-pow 4207

This theorem depends on definitions:  df-bi 117  df-dc 836  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-rab 2484  df-v 2765  df-dif 3159  df-in 3163  df-ss 3170  df-nul 3451  df-pw 3607  df-sn 3628  df-exmid 4228

This theorem is referenced by:  exmidel  4238