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Theorem dcextest 4463
Description: If it is decidable whether {𝑥𝜑} is a set, then ¬ 𝜑 is decidable (where 𝑥 does not occur in 𝜑). From this fact, we can deduce (outside the formal system, since we cannot quantify over classes) that if it is decidable whether any class is a set, then "weak excluded middle" (that is, any negated proposition ¬ 𝜑 is decidable) holds. (Contributed by Jim Kingdon, 3-Jul-2022.)
Hypothesis
Ref Expression
dcextest.ex DECID {𝑥𝜑} ∈ V
Assertion
Ref Expression
dcextest DECID ¬ 𝜑
Distinct variable group:   𝜑,𝑥

Proof of Theorem dcextest
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dcextest.ex . . . 4 DECID {𝑥𝜑} ∈ V
2 exmiddc 804 . . . 4 (DECID {𝑥𝜑} ∈ V → ({𝑥𝜑} ∈ V ∨ ¬ {𝑥𝜑} ∈ V))
31, 2ax-mp 5 . . 3 ({𝑥𝜑} ∈ V ∨ ¬ {𝑥𝜑} ∈ V)
4 vprc 4028 . . . . . . 7 ¬ V ∈ V
5 df-v 2660 . . . . . . . . 9 V = {𝑥𝑥 = 𝑥}
6 equid 1660 . . . . . . . . . . 11 𝑥 = 𝑥
7 pm5.1im 172 . . . . . . . . . . 11 (𝑥 = 𝑥 → (𝜑 → (𝑥 = 𝑥𝜑)))
86, 7ax-mp 5 . . . . . . . . . 10 (𝜑 → (𝑥 = 𝑥𝜑))
98abbidv 2233 . . . . . . . . 9 (𝜑 → {𝑥𝑥 = 𝑥} = {𝑥𝜑})
105, 9syl5req 2161 . . . . . . . 8 (𝜑 → {𝑥𝜑} = V)
1110eleq1d 2184 . . . . . . 7 (𝜑 → ({𝑥𝜑} ∈ V ↔ V ∈ V))
124, 11mtbiri 647 . . . . . 6 (𝜑 → ¬ {𝑥𝜑} ∈ V)
1312con2i 599 . . . . 5 ({𝑥𝜑} ∈ V → ¬ 𝜑)
14 vex 2661 . . . . . . . . . 10 𝑦 ∈ V
15 biidd 171 . . . . . . . . . 10 (𝑥 = 𝑦 → (𝜑𝜑))
1614, 15elab 2800 . . . . . . . . 9 (𝑦 ∈ {𝑥𝜑} ↔ 𝜑)
1716notbii 640 . . . . . . . 8 𝑦 ∈ {𝑥𝜑} ↔ ¬ 𝜑)
1817biimpri 132 . . . . . . 7 𝜑 → ¬ 𝑦 ∈ {𝑥𝜑})
1918eq0rdv 3375 . . . . . 6 𝜑 → {𝑥𝜑} = ∅)
20 0ex 4023 . . . . . 6 ∅ ∈ V
2119, 20syl6eqel 2206 . . . . 5 𝜑 → {𝑥𝜑} ∈ V)
2213, 21impbii 125 . . . 4 ({𝑥𝜑} ∈ V ↔ ¬ 𝜑)
2322notbii 640 . . . 4 (¬ {𝑥𝜑} ∈ V ↔ ¬ ¬ 𝜑)
2422, 23orbi12i 736 . . 3 (({𝑥𝜑} ∈ V ∨ ¬ {𝑥𝜑} ∈ V) ↔ (¬ 𝜑 ∨ ¬ ¬ 𝜑))
253, 24mpbi 144 . 2 𝜑 ∨ ¬ ¬ 𝜑)
26 df-dc 803 . 2 (DECID ¬ 𝜑 ↔ (¬ 𝜑 ∨ ¬ ¬ 𝜑))
2725, 26mpbir 145 1 DECID ¬ 𝜑
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 104  wo 680  DECID wdc 802  wcel 1463  {cab 2101  Vcvv 2658  c0 3331
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-nul 4022
This theorem depends on definitions:  df-bi 116  df-dc 803  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-dif 3041  df-in 3045  df-ss 3052  df-nul 3332
This theorem is referenced by: (None)
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