ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  rabrsndc GIF version

Theorem rabrsndc 3591
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 822 . . . . . . . 8 (DECID 𝜑 → (¬ 𝜑𝜑))
42, 3ax-mp 5 . . . . . . 7 𝜑𝜑)
54sbcth 2922 . . . . . 6 (𝐴 ∈ V → [𝐴 / 𝑥]𝜑𝜑))
61, 5ax-mp 5 . . . . 5 [𝐴 / 𝑥]𝜑𝜑)
7 sbcor 2953 . . . . 5 ([𝐴 / 𝑥]𝜑𝜑) ↔ ([𝐴 / 𝑥] ¬ 𝜑[𝐴 / 𝑥]𝜑))
86, 7mpbi 144 . . . 4 ([𝐴 / 𝑥] ¬ 𝜑[𝐴 / 𝑥]𝜑)
9 ralsns 3562 . . . . . 6 (𝐴 ∈ V → (∀𝑥 ∈ {𝐴} ¬ 𝜑[𝐴 / 𝑥] ¬ 𝜑))
101, 9ax-mp 5 . . . . 5 (∀𝑥 ∈ {𝐴} ¬ 𝜑[𝐴 / 𝑥] ¬ 𝜑)
11 ralsns 3562 . . . . . 6 (𝐴 ∈ V → (∀𝑥 ∈ {𝐴}𝜑[𝐴 / 𝑥]𝜑))
121, 11ax-mp 5 . . . . 5 (∀𝑥 ∈ {𝐴}𝜑[𝐴 / 𝑥]𝜑)
1310, 12orbi12i 753 . . . 4 ((∀𝑥 ∈ {𝐴} ¬ 𝜑 ∨ ∀𝑥 ∈ {𝐴}𝜑) ↔ ([𝐴 / 𝑥] ¬ 𝜑[𝐴 / 𝑥]𝜑))
148, 13mpbir 145 . . 3 (∀𝑥 ∈ {𝐴} ¬ 𝜑 ∨ ∀𝑥 ∈ {𝐴}𝜑)
15 rabeq0 3392 . . . 4 ({𝑥 ∈ {𝐴} ∣ 𝜑} = ∅ ↔ ∀𝑥 ∈ {𝐴} ¬ 𝜑)
16 eqcom 2141 . . . . 5 ({𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴} ↔ {𝐴} = {𝑥 ∈ {𝐴} ∣ 𝜑})
17 rabid2 2607 . . . . 5 ({𝐴} = {𝑥 ∈ {𝐴} ∣ 𝜑} ↔ ∀𝑥 ∈ {𝐴}𝜑)
1816, 17bitri 183 . . . 4 ({𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴} ↔ ∀𝑥 ∈ {𝐴}𝜑)
1915, 18orbi12i 753 . . 3 (({𝑥 ∈ {𝐴} ∣ 𝜑} = ∅ ∨ {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴}) ↔ (∀𝑥 ∈ {𝐴} ¬ 𝜑 ∨ ∀𝑥 ∈ {𝐴}𝜑))
2014, 19mpbir 145 . 2 ({𝑥 ∈ {𝐴} ∣ 𝜑} = ∅ ∨ {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴})
21 eqeq1 2146 . . 3 (𝑀 = {𝑥 ∈ {𝐴} ∣ 𝜑} → (𝑀 = ∅ ↔ {𝑥 ∈ {𝐴} ∣ 𝜑} = ∅))
22 eqeq1 2146 . . 3 (𝑀 = {𝑥 ∈ {𝐴} ∣ 𝜑} → (𝑀 = {𝐴} ↔ {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴}))
2321, 22orbi12d 782 . 2 (𝑀 = {𝑥 ∈ {𝐴} ∣ 𝜑} → ((𝑀 = ∅ ∨ 𝑀 = {𝐴}) ↔ ({𝑥 ∈ {𝐴} ∣ 𝜑} = ∅ ∨ {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴})))
2420, 23mpbiri 167 1 (𝑀 = {𝑥 ∈ {𝐴} ∣ 𝜑} → (𝑀 = ∅ ∨ 𝑀 = {𝐴}))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 104  wo 697  DECID wdc 819   = wceq 1331  wcel 1480  wral 2416  {crab 2420  Vcvv 2686  [wsbc 2909  c0 3363  {csn 3527
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-dc 820  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-nul 3364  df-sn 3533
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator