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Theorem rabrsndc 3412
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 736 . . . . . . . 8 (DECID φ → (¬ φ φ))
42, 3ax-mp 7 . . . . . . 7 φ φ)
54sbcth 2754 . . . . . 6 (A V → [A / x]φ φ))
61, 5ax-mp 7 . . . . 5 [A / x]φ φ)
7 sbcor 2784 . . . . 5 ([A / x]φ φ) ↔ ([A / x] ¬ φ [A / x]φ))
86, 7mpbi 133 . . . 4 ([A / x] ¬ φ [A / x]φ)
9 ralsns 3382 . . . . . 6 (A V → (x {A} ¬ φ[A / x] ¬ φ))
101, 9ax-mp 7 . . . . 5 (x {A} ¬ φ[A / x] ¬ φ)
11 ralsns 3382 . . . . . 6 (A V → (x {A}φ[A / x]φ))
121, 11ax-mp 7 . . . . 5 (x {A}φ[A / x]φ)
1310, 12orbi12i 668 . . . 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 3224 . . . 4 ({x {A} ∣ φ} = ∅ ↔ x {A} ¬ φ)
16 eqcom 2024 . . . . 5 ({x {A} ∣ φ} = {A} ↔ {A} = {x {A} ∣ φ})
17 rabid2 2464 . . . . 5 ({A} = {x {A} ∣ φ} ↔ x {A}φ)
1816, 17bitri 173 . . . 4 ({x {A} ∣ φ} = {A} ↔ x {A}φ)
1915, 18orbi12i 668 . . 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 2028 . . 3 (𝑀 = {x {A} ∣ φ} → (𝑀 = ∅ ↔ {x {A} ∣ φ} = ∅))
22 eqeq1 2028 . . 3 (𝑀 = {x {A} ∣ φ} → (𝑀 = {A} ↔ {x {A} ∣ φ} = {A}))
2321, 22orbi12d 694 . 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 616  DECID wdc 733   = wceq 1228   wcel 1374  wral 2284  {crab 2288  Vcvv 2535  [wsbc 2741  c0 3201  {csn 3350
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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-dc 734  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rab 2293  df-v 2537  df-sbc 2742  df-dif 2897  df-nul 3202  df-sn 3356
This theorem is referenced by: (None)
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