Theorem exmidel 4257

Description: Excluded middle is equivalent to decidability of membership for two arbitrary sets. (Contributed by Jim Kingdon, 18-Jun-2022.)

Assertion

Ref	Expression
exmidel	⊢ (EXMID ↔ ∀𝑥∀𝑦DECID 𝑥 ∈ 𝑦)

Distinct variable group:   𝑥,𝑦

Proof of Theorem exmidel

Step	Hyp	Ref	Expression
1		exmidexmid 4248	. . 3 ⊢ (EXMID → DECID 𝑥 ∈ 𝑦)
2	1	alrimivv 1899	. 2 ⊢ (EXMID → ∀𝑥∀𝑦DECID 𝑥 ∈ 𝑦)
3		0ex 4179	. . . 4 ⊢ ∅ ∈ V
4		eleq1 2269	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 ∈ 𝑦 ↔ ∅ ∈ 𝑦))
5	4	dcbid 840	. . . . 5 ⊢ (𝑥 = ∅ → (DECID 𝑥 ∈ 𝑦 ↔ DECID ∅ ∈ 𝑦))
6	5	albidv 1848	. . . 4 ⊢ (𝑥 = ∅ → (∀𝑦DECID 𝑥 ∈ 𝑦 ↔ ∀𝑦DECID ∅ ∈ 𝑦))
7	3, 6	spcv 2871	. . 3 ⊢ (∀𝑥∀𝑦DECID 𝑥 ∈ 𝑦 → ∀𝑦DECID ∅ ∈ 𝑦)
8		exmid0el 4256	. . 3 ⊢ (EXMID ↔ ∀𝑦DECID ∅ ∈ 𝑦)
9	7, 8	sylibr 134	. 2 ⊢ (∀𝑥∀𝑦DECID 𝑥 ∈ 𝑦 → EXMID)
10	2, 9	impbii 126	1 ⊢ (EXMID ↔ ∀𝑥∀𝑦DECID 𝑥 ∈ 𝑦)

Syntax hints:   ↔ wb 105  DECID wdc 836  ∀wal 1371   = wceq 1373   ∈ wcel 2177  ∅c0 3464  EXMIDwem 4246

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-nul 4178  ax-pow 4226

This theorem depends on definitions:  df-bi 117  df-dc 837  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-rab 2494  df-v 2775  df-dif 3172  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-exmid 4247

This theorem is referenced by: (None)