Theorem exmidel 4161

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 4152	. . 3 ⊢ (EXMID → DECID 𝑥 ∈ 𝑦)
2	1	alrimivv 1852	. 2 ⊢ (EXMID → ∀𝑥∀𝑦DECID 𝑥 ∈ 𝑦)
3		0ex 4087	. . . 4 ⊢ ∅ ∈ V
4		eleq1 2217	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 ∈ 𝑦 ↔ ∅ ∈ 𝑦))
5	4	dcbid 824	. . . . 5 ⊢ (𝑥 = ∅ → (DECID 𝑥 ∈ 𝑦 ↔ DECID ∅ ∈ 𝑦))
6	5	albidv 1801	. . . 4 ⊢ (𝑥 = ∅ → (∀𝑦DECID 𝑥 ∈ 𝑦 ↔ ∀𝑦DECID ∅ ∈ 𝑦))
7	3, 6	spcv 2803	. . 3 ⊢ (∀𝑥∀𝑦DECID 𝑥 ∈ 𝑦 → ∀𝑦DECID ∅ ∈ 𝑦)
8		exmid0el 4160	. . 3 ⊢ (EXMID ↔ ∀𝑦DECID ∅ ∈ 𝑦)
9	7, 8	sylibr 133	. 2 ⊢ (∀𝑥∀𝑦DECID 𝑥 ∈ 𝑦 → EXMID)
10	2, 9	impbii 125	1 ⊢ (EXMID ↔ ∀𝑥∀𝑦DECID 𝑥 ∈ 𝑦)

Syntax hints:   ↔ wb 104  DECID wdc 820  ∀wal 1330   = wceq 1332   ∈ wcel 2125  ∅c0 3390  EXMIDwem 4150

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-nul 4086  ax-pow 4130

This theorem depends on definitions:  df-bi 116  df-dc 821  df-tru 1335  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-rab 2441  df-v 2711  df-dif 3100  df-in 3104  df-ss 3111  df-nul 3391  df-pw 3541  df-sn 3562  df-exmid 4151

This theorem is referenced by: (None)