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Theorem rabrsndc 3686
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 838 . . . . . . . 8 (DECID 𝜑 → (¬ 𝜑𝜑))
42, 3ax-mp 5 . . . . . . 7 𝜑𝜑)
54sbcth 2999 . . . . . 6 (𝐴 ∈ V → [𝐴 / 𝑥]𝜑𝜑))
61, 5ax-mp 5 . . . . 5 [𝐴 / 𝑥]𝜑𝜑)
7 sbcor 3030 . . . . 5 ([𝐴 / 𝑥]𝜑𝜑) ↔ ([𝐴 / 𝑥] ¬ 𝜑[𝐴 / 𝑥]𝜑))
86, 7mpbi 145 . . . 4 ([𝐴 / 𝑥] ¬ 𝜑[𝐴 / 𝑥]𝜑)
9 ralsns 3656 . . . . . 6 (𝐴 ∈ V → (∀𝑥 ∈ {𝐴} ¬ 𝜑[𝐴 / 𝑥] ¬ 𝜑))
101, 9ax-mp 5 . . . . 5 (∀𝑥 ∈ {𝐴} ¬ 𝜑[𝐴 / 𝑥] ¬ 𝜑)
11 ralsns 3656 . . . . . 6 (𝐴 ∈ V → (∀𝑥 ∈ {𝐴}𝜑[𝐴 / 𝑥]𝜑))
121, 11ax-mp 5 . . . . 5 (∀𝑥 ∈ {𝐴}𝜑[𝐴 / 𝑥]𝜑)
1310, 12orbi12i 765 . . . 4 ((∀𝑥 ∈ {𝐴} ¬ 𝜑 ∨ ∀𝑥 ∈ {𝐴}𝜑) ↔ ([𝐴 / 𝑥] ¬ 𝜑[𝐴 / 𝑥]𝜑))
148, 13mpbir 146 . . 3 (∀𝑥 ∈ {𝐴} ¬ 𝜑 ∨ ∀𝑥 ∈ {𝐴}𝜑)
15 rabeq0 3476 . . . 4 ({𝑥 ∈ {𝐴} ∣ 𝜑} = ∅ ↔ ∀𝑥 ∈ {𝐴} ¬ 𝜑)
16 eqcom 2195 . . . . 5 ({𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴} ↔ {𝐴} = {𝑥 ∈ {𝐴} ∣ 𝜑})
17 rabid2 2671 . . . . 5 ({𝐴} = {𝑥 ∈ {𝐴} ∣ 𝜑} ↔ ∀𝑥 ∈ {𝐴}𝜑)
1816, 17bitri 184 . . . 4 ({𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴} ↔ ∀𝑥 ∈ {𝐴}𝜑)
1915, 18orbi12i 765 . . 3 (({𝑥 ∈ {𝐴} ∣ 𝜑} = ∅ ∨ {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴}) ↔ (∀𝑥 ∈ {𝐴} ¬ 𝜑 ∨ ∀𝑥 ∈ {𝐴}𝜑))
2014, 19mpbir 146 . 2 ({𝑥 ∈ {𝐴} ∣ 𝜑} = ∅ ∨ {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴})
21 eqeq1 2200 . . 3 (𝑀 = {𝑥 ∈ {𝐴} ∣ 𝜑} → (𝑀 = ∅ ↔ {𝑥 ∈ {𝐴} ∣ 𝜑} = ∅))
22 eqeq1 2200 . . 3 (𝑀 = {𝑥 ∈ {𝐴} ∣ 𝜑} → (𝑀 = {𝐴} ↔ {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴}))
2321, 22orbi12d 794 . 2 (𝑀 = {𝑥 ∈ {𝐴} ∣ 𝜑} → ((𝑀 = ∅ ∨ 𝑀 = {𝐴}) ↔ ({𝑥 ∈ {𝐴} ∣ 𝜑} = ∅ ∨ {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴})))
2420, 23mpbiri 168 1 (𝑀 = {𝑥 ∈ {𝐴} ∣ 𝜑} → (𝑀 = ∅ ∨ 𝑀 = {𝐴}))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 105  wo 709  DECID wdc 835   = wceq 1364  wcel 2164  wral 2472  {crab 2476  Vcvv 2760  [wsbc 2985  c0 3446  {csn 3618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175
This theorem depends on definitions:  df-bi 117  df-dc 836  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rab 2481  df-v 2762  df-sbc 2986  df-dif 3155  df-nul 3447  df-sn 3624
This theorem is referenced by: (None)
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