Theorem pw1dc0el 6877

Description: Another equivalent of excluded middle, which is a mere reformulation of the definition. (Contributed by BJ, 9-Aug-2024.)

Assertion

Ref	Expression
pw1dc0el	⊢ (EXMID ↔ ∀𝑥 ∈ 𝒫 1oDECID ∅ ∈ 𝑥)

Proof of Theorem pw1dc0el

Step	Hyp	Ref	Expression
1		df1o2 6397	. . . . . . 7 ⊢ 1o = {∅}
2	1	eqcomi 2169	. . . . . 6 ⊢ {∅} = 1o
3	2	sseq2i 3169	. . . . 5 ⊢ (𝑥 ⊆ {∅} ↔ 𝑥 ⊆ 1o)
4		velpw 3566	. . . . 5 ⊢ (𝑥 ∈ 𝒫 1o ↔ 𝑥 ⊆ 1o)
5	3, 4	bitr4i 186	. . . 4 ⊢ (𝑥 ⊆ {∅} ↔ 𝑥 ∈ 𝒫 1o)
6	5	imbi1i 237	. . 3 ⊢ ((𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥) ↔ (𝑥 ∈ 𝒫 1o → DECID ∅ ∈ 𝑥))
7	6	albii 1458	. 2 ⊢ (∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥) ↔ ∀𝑥(𝑥 ∈ 𝒫 1o → DECID ∅ ∈ 𝑥))
8		df-exmid 4174	. 2 ⊢ (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥))
9		df-ral 2449	. 2 ⊢ (∀𝑥 ∈ 𝒫 1oDECID ∅ ∈ 𝑥 ↔ ∀𝑥(𝑥 ∈ 𝒫 1o → DECID ∅ ∈ 𝑥))
10	7, 8, 9	3bitr4i 211	1 ⊢ (EXMID ↔ ∀𝑥 ∈ 𝒫 1oDECID ∅ ∈ 𝑥)

Syntax hints:   → wi 4   ↔ wb 104  DECID wdc 824  ∀wal 1341   ∈ wcel 2136  ∀wral 2444   ⊆ wss 3116  ∅c0 3409  𝒫 cpw 3559  {csn 3576  EXMIDwem 4173  1_oc1o 6377

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-exmid 4174  df-suc 4349  df-1o 6384

This theorem is referenced by:  pw1dc1  6879