Theorem pw1dc0el 7173

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 6663	. . . . . . 7 ⊢ 1o = {∅}
2	1	eqcomi 2238	. . . . . 6 ⊢ {∅} = 1o
3	2	sseq2i 3267	. . . . 5 ⊢ (𝑥 ⊆ {∅} ↔ 𝑥 ⊆ 1o)
4		velpw 3678	. . . . 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 1519	. 2 ⊢ (∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥) ↔ ∀𝑥(𝑥 ∈ 𝒫 1o → DECID ∅ ∈ 𝑥))
8		df-exmid 4310	. 2 ⊢ (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥))
9		df-ral 2527	. 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 842  ∀wal 1396   ∈ wcel 2205  ∀wral 2522   ⊆ wss 3213  ∅c0 3510  𝒫 cpw 3671  {csn 3691  EXMIDwem 4309  1_oc1o 6642

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-v 2817  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-pw 3673  df-exmid 4310  df-suc 4494  df-1o 6649

This theorem is referenced by:  pw1dc1  7176