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Theorem rabrsndc 3510
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 783 . . . . . . . 8 (DECID 𝜑 → (¬ 𝜑𝜑))
42, 3ax-mp 7 . . . . . . 7 𝜑𝜑)
54sbcth 2853 . . . . . 6 (𝐴 ∈ V → [𝐴 / 𝑥]𝜑𝜑))
61, 5ax-mp 7 . . . . 5 [𝐴 / 𝑥]𝜑𝜑)
7 sbcor 2883 . . . . 5 ([𝐴 / 𝑥]𝜑𝜑) ↔ ([𝐴 / 𝑥] ¬ 𝜑[𝐴 / 𝑥]𝜑))
86, 7mpbi 143 . . . 4 ([𝐴 / 𝑥] ¬ 𝜑[𝐴 / 𝑥]𝜑)
9 ralsns 3481 . . . . . 6 (𝐴 ∈ V → (∀𝑥 ∈ {𝐴} ¬ 𝜑[𝐴 / 𝑥] ¬ 𝜑))
101, 9ax-mp 7 . . . . 5 (∀𝑥 ∈ {𝐴} ¬ 𝜑[𝐴 / 𝑥] ¬ 𝜑)
11 ralsns 3481 . . . . . 6 (𝐴 ∈ V → (∀𝑥 ∈ {𝐴}𝜑[𝐴 / 𝑥]𝜑))
121, 11ax-mp 7 . . . . 5 (∀𝑥 ∈ {𝐴}𝜑[𝐴 / 𝑥]𝜑)
1310, 12orbi12i 716 . . . 4 ((∀𝑥 ∈ {𝐴} ¬ 𝜑 ∨ ∀𝑥 ∈ {𝐴}𝜑) ↔ ([𝐴 / 𝑥] ¬ 𝜑[𝐴 / 𝑥]𝜑))
148, 13mpbir 144 . . 3 (∀𝑥 ∈ {𝐴} ¬ 𝜑 ∨ ∀𝑥 ∈ {𝐴}𝜑)
15 rabeq0 3312 . . . 4 ({𝑥 ∈ {𝐴} ∣ 𝜑} = ∅ ↔ ∀𝑥 ∈ {𝐴} ¬ 𝜑)
16 eqcom 2090 . . . . 5 ({𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴} ↔ {𝐴} = {𝑥 ∈ {𝐴} ∣ 𝜑})
17 rabid2 2543 . . . . 5 ({𝐴} = {𝑥 ∈ {𝐴} ∣ 𝜑} ↔ ∀𝑥 ∈ {𝐴}𝜑)
1816, 17bitri 182 . . . 4 ({𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴} ↔ ∀𝑥 ∈ {𝐴}𝜑)
1915, 18orbi12i 716 . . 3 (({𝑥 ∈ {𝐴} ∣ 𝜑} = ∅ ∨ {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴}) ↔ (∀𝑥 ∈ {𝐴} ¬ 𝜑 ∨ ∀𝑥 ∈ {𝐴}𝜑))
2014, 19mpbir 144 . 2 ({𝑥 ∈ {𝐴} ∣ 𝜑} = ∅ ∨ {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴})
21 eqeq1 2094 . . 3 (𝑀 = {𝑥 ∈ {𝐴} ∣ 𝜑} → (𝑀 = ∅ ↔ {𝑥 ∈ {𝐴} ∣ 𝜑} = ∅))
22 eqeq1 2094 . . 3 (𝑀 = {𝑥 ∈ {𝐴} ∣ 𝜑} → (𝑀 = {𝐴} ↔ {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴}))
2321, 22orbi12d 742 . 2 (𝑀 = {𝑥 ∈ {𝐴} ∣ 𝜑} → ((𝑀 = ∅ ∨ 𝑀 = {𝐴}) ↔ ({𝑥 ∈ {𝐴} ∣ 𝜑} = ∅ ∨ {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴})))
2420, 23mpbiri 166 1 (𝑀 = {𝑥 ∈ {𝐴} ∣ 𝜑} → (𝑀 = ∅ ∨ 𝑀 = {𝐴}))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 103  wo 664  DECID wdc 780   = wceq 1289  wcel 1438  wral 2359  {crab 2363  Vcvv 2619  [wsbc 2840  c0 3286  {csn 3446
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-dc 781  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rab 2368  df-v 2621  df-sbc 2841  df-dif 3001  df-nul 3287  df-sn 3452
This theorem is referenced by: (None)
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