Theorem exmidel 4128

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 4120	. . 3 ⊢ (EXMID → DECID 𝑥 ∈ 𝑦)
2	1	alrimivv 1847	. 2 ⊢ (EXMID → ∀𝑥∀𝑦DECID 𝑥 ∈ 𝑦)
3		0ex 4055	. . . 4 ⊢ ∅ ∈ V
4		eleq1 2202	. . . . . 6 ⊢ (𝑥 = ∅ → (𝑥 ∈ 𝑦 ↔ ∅ ∈ 𝑦))
5	4	dcbid 823	. . . . 5 ⊢ (𝑥 = ∅ → (DECID 𝑥 ∈ 𝑦 ↔ DECID ∅ ∈ 𝑦))
6	5	albidv 1796	. . . 4 ⊢ (𝑥 = ∅ → (∀𝑦DECID 𝑥 ∈ 𝑦 ↔ ∀𝑦DECID ∅ ∈ 𝑦))
7	3, 6	spcv 2779	. . 3 ⊢ (∀𝑥∀𝑦DECID 𝑥 ∈ 𝑦 → ∀𝑦DECID ∅ ∈ 𝑦)
8		exmid0el 4127	. . 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 819  ∀wal 1329   = wceq 1331   ∈ wcel 1480  ∅c0 3363  EXMIDwem 4118

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098

This theorem depends on definitions:  df-bi 116  df-dc 820  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rab 2425  df-v 2688  df-dif 3073  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-exmid 4119

This theorem is referenced by: (None)