Theorem pw1dc0el 7041

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 6545	. . . . . . 7 ⊢ 1o = {∅}
2	1	eqcomi 2213	. . . . . 6 ⊢ {∅} = 1o
3	2	sseq2i 3231	. . . . 5 ⊢ (𝑥 ⊆ {∅} ↔ 𝑥 ⊆ 1o)
4		velpw 3636	. . . . 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 1496	. 2 ⊢ (∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥) ↔ ∀𝑥(𝑥 ∈ 𝒫 1o → DECID ∅ ∈ 𝑥))
8		df-exmid 4258	. 2 ⊢ (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥))
9		df-ral 2493	. 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 838  ∀wal 1373   ∈ wcel 2180  ∀wral 2488   ⊆ wss 3177  ∅c0 3471  𝒫 cpw 3629  {csn 3646  EXMIDwem 4257  1_oc1o 6525

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 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-ext 2191

This theorem depends on definitions:  df-bi 117  df-tru 1378  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ral 2493  df-v 2781  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-nul 3472  df-pw 3631  df-exmid 4258  df-suc 4439  df-1o 6532

This theorem is referenced by:  pw1dc1  7044