Theorem pw1dc0el 7212

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 6695	. . . . . . 7 ⊢ 1o = {∅}
2	1	eqcomi 2242	. . . . . 6 ⊢ {∅} = 1o
3	2	sseq2i 3275	. . . . 5 ⊢ (𝑥 ⊆ {∅} ↔ 𝑥 ⊆ 1o)
4		velpw 3695	. . . . 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 1523	. 2 ⊢ (∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥) ↔ ∀𝑥(𝑥 ∈ 𝒫 1o → DECID ∅ ∈ 𝑥))
8		df-exmid 4330	. 2 ⊢ (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥))
9		df-ral 2533	. 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 846  ∀wal 1400   ∈ wcel 2209  ∀wral 2528   ⊆ wss 3220  ∅c0 3520  𝒫 cpw 3688  {csn 3708  EXMIDwem 4329  1_oc1o 6674

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-exmid 4330  df-suc 4514  df-1o 6681

This theorem is referenced by:  pw1dc1  7215