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Theorem pw1dc0el 6906
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 6425 . . . . . . 7 1o = {∅}
21eqcomi 2181 . . . . . 6 {∅} = 1o
32sseq2i 3182 . . . . 5 (𝑥 ⊆ {∅} ↔ 𝑥 ⊆ 1o)
4 velpw 3582 . . . . 5 (𝑥 ∈ 𝒫 1o𝑥 ⊆ 1o)
53, 4bitr4i 187 . . . 4 (𝑥 ⊆ {∅} ↔ 𝑥 ∈ 𝒫 1o)
65imbi1i 238 . . 3 ((𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥) ↔ (𝑥 ∈ 𝒫 1oDECID ∅ ∈ 𝑥))
76albii 1470 . 2 (∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥) ↔ ∀𝑥(𝑥 ∈ 𝒫 1oDECID ∅ ∈ 𝑥))
8 df-exmid 4193 . 2 (EXMID ↔ ∀𝑥(𝑥 ⊆ {∅} → DECID ∅ ∈ 𝑥))
9 df-ral 2460 . 2 (∀𝑥 ∈ 𝒫 1oDECID ∅ ∈ 𝑥 ↔ ∀𝑥(𝑥 ∈ 𝒫 1oDECID ∅ ∈ 𝑥))
107, 8, 93bitr4i 212 1 (EXMID ↔ ∀𝑥 ∈ 𝒫 1oDECID ∅ ∈ 𝑥)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  DECID wdc 834  wal 1351  wcel 2148  wral 2455  wss 3129  c0 3422  𝒫 cpw 3575  {csn 3592  EXMIDwem 4192  1oc1o 6405
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-exmid 4193  df-suc 4369  df-1o 6412
This theorem is referenced by:  pw1dc1  6908
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