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Theorem pw1dc0el 6849
 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
StepHypRef Expression
1 df1o2 6370 . . . . . . 7 1o = {∅}
21eqcomi 2161 . . . . . 6 {∅} = 1o
32sseq2i 3155 . . . . 5 (𝑥 ⊆ {∅} ↔ 𝑥 ⊆ 1o)
4 velpw 3550 . . . . 5 (𝑥 ∈ 𝒫 1o𝑥 ⊆ 1o)
53, 4bitr4i 186 . . . 4 (𝑥 ⊆ {∅} ↔ 𝑥 ∈ 𝒫 1o)
65imbi1i 237 . . 3 ((𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥) ↔ (𝑥 ∈ 𝒫 1oDECID ∅ ∈ 𝑥))
76albii 1450 . 2 (∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥) ↔ ∀𝑥(𝑥 ∈ 𝒫 1oDECID ∅ ∈ 𝑥))
8 df-exmid 4155 . 2 (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥))
9 df-ral 2440 . 2 (∀𝑥 ∈ 𝒫 1oDECID ∅ ∈ 𝑥 ↔ ∀𝑥(𝑥 ∈ 𝒫 1oDECID ∅ ∈ 𝑥))
107, 8, 93bitr4i 211 1 (EXMID ↔ ∀𝑥 ∈ 𝒫 1oDECID ∅ ∈ 𝑥)
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104  DECID wdc 820  ∀wal 1333   ∈ wcel 2128  ∀wral 2435   ⊆ wss 3102  ∅c0 3394  𝒫 cpw 3543  {csn 3560  EXMIDwem 4154  1oc1o 6350 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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-exmid 4155  df-suc 4330  df-1o 6357 This theorem is referenced by:  pw1dc1  6851
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