Theorem pw1dc0el 6924

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 6443	. . . . . . 7 ⊢ 1o = {∅}
2	1	eqcomi 2191	. . . . . 6 ⊢ {∅} = 1o
3	2	sseq2i 3194	. . . . 5 ⊢ (𝑥 ⊆ {∅} ↔ 𝑥 ⊆ 1o)
4		velpw 3594	. . . . 5 ⊢ (𝑥 ∈ 𝒫 1o ↔ 𝑥 ⊆ 1o)
5	3, 4	bitr4i 187	. . . 4 ⊢ (𝑥 ⊆ {∅} ↔ 𝑥 ∈ 𝒫 1o)
6	5	imbi1i 238	. . 3 ⊢ ((𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥) ↔ (𝑥 ∈ 𝒫 1o → DECID ∅ ∈ 𝑥))
7	6	albii 1480	. 2 ⊢ (∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥) ↔ ∀𝑥(𝑥 ∈ 𝒫 1o → DECID ∅ ∈ 𝑥))
8		df-exmid 4207	. 2 ⊢ (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥))
9		df-ral 2470	. 2 ⊢ (∀𝑥 ∈ 𝒫 1oDECID ∅ ∈ 𝑥 ↔ ∀𝑥(𝑥 ∈ 𝒫 1o → DECID ∅ ∈ 𝑥))
10	7, 8, 9	3bitr4i 212	1 ⊢ (EXMID ↔ ∀𝑥 ∈ 𝒫 1oDECID ∅ ∈ 𝑥)

Syntax hints:   → wi 4   ↔ wb 105  DECID wdc 835  ∀wal 1361   ∈ wcel 2158  ∀wral 2465   ⊆ wss 3141  ∅c0 3434  𝒫 cpw 3587  {csn 3604  EXMIDwem 4206  1_oc1o 6423

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-v 2751  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-exmid 4207  df-suc 4383  df-1o 6430

This theorem is referenced by:  pw1dc1  6926