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Theorem rabrsndc 3429
Description: A class abstraction over a decidable proposition restricted to a singleton is either the empty set or the singleton itself. (Contributed by Jim Kingdon, 8-Aug-2018.)
Hypotheses
Ref Expression
rabrsndc.1 A V
rabrsndc.2 DECID φ
Assertion
Ref Expression
rabrsndc (𝑀 = {x {A} ∣ φ} → (𝑀 = ∅ 𝑀 = {A}))
Distinct variable group:   x,A
Allowed substitution hints:   φ(x)   𝑀(x)

Proof of Theorem rabrsndc
StepHypRef Expression
1 rabrsndc.1 . . . . . 6 A V
2 rabrsndc.2 . . . . . . . 8 DECID φ
3 pm2.1dc 744 . . . . . . . 8 (DECID φ → (¬ φ φ))
42, 3ax-mp 7 . . . . . . 7 φ φ)
54sbcth 2771 . . . . . 6 (A V → [A / x]φ φ))
61, 5ax-mp 7 . . . . 5 [A / x]φ φ)
7 sbcor 2801 . . . . 5 ([A / x]φ φ) ↔ ([A / x] ¬ φ [A / x]φ))
86, 7mpbi 133 . . . 4 ([A / x] ¬ φ [A / x]φ)
9 ralsns 3399 . . . . . 6 (A V → (x {A} ¬ φ[A / x] ¬ φ))
101, 9ax-mp 7 . . . . 5 (x {A} ¬ φ[A / x] ¬ φ)
11 ralsns 3399 . . . . . 6 (A V → (x {A}φ[A / x]φ))
121, 11ax-mp 7 . . . . 5 (x {A}φ[A / x]φ)
1310, 12orbi12i 680 . . . 4 ((x {A} ¬ φ x {A}φ) ↔ ([A / x] ¬ φ [A / x]φ))
148, 13mpbir 134 . . 3 (x {A} ¬ φ x {A}φ)
15 rabeq0 3241 . . . 4 ({x {A} ∣ φ} = ∅ ↔ x {A} ¬ φ)
16 eqcom 2039 . . . . 5 ({x {A} ∣ φ} = {A} ↔ {A} = {x {A} ∣ φ})
17 rabid2 2480 . . . . 5 ({A} = {x {A} ∣ φ} ↔ x {A}φ)
1816, 17bitri 173 . . . 4 ({x {A} ∣ φ} = {A} ↔ x {A}φ)
1915, 18orbi12i 680 . . 3 (({x {A} ∣ φ} = ∅ {x {A} ∣ φ} = {A}) ↔ (x {A} ¬ φ x {A}φ))
2014, 19mpbir 134 . 2 ({x {A} ∣ φ} = ∅ {x {A} ∣ φ} = {A})
21 eqeq1 2043 . . 3 (𝑀 = {x {A} ∣ φ} → (𝑀 = ∅ ↔ {x {A} ∣ φ} = ∅))
22 eqeq1 2043 . . 3 (𝑀 = {x {A} ∣ φ} → (𝑀 = {A} ↔ {x {A} ∣ φ} = {A}))
2321, 22orbi12d 706 . 2 (𝑀 = {x {A} ∣ φ} → ((𝑀 = ∅ 𝑀 = {A}) ↔ ({x {A} ∣ φ} = ∅ {x {A} ∣ φ} = {A})))
2420, 23mpbiri 157 1 (𝑀 = {x {A} ∣ φ} → (𝑀 = ∅ 𝑀 = {A}))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 98   wo 628  DECID wdc 741   = wceq 1242   wcel 1390  wral 2300  {crab 2304  Vcvv 2551  [wsbc 2758  c0 3218  {csn 3367
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-dc 742  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rab 2309  df-v 2553  df-sbc 2759  df-dif 2914  df-nul 3219  df-sn 3373
This theorem is referenced by: (None)
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