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Theorem rabrsndc 3438
 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 745 . . . . . . . 8 (DECID 𝜑 → (¬ 𝜑𝜑))
42, 3ax-mp 7 . . . . . . 7 𝜑𝜑)
54sbcth 2777 . . . . . 6 (𝐴 ∈ V → [𝐴 / 𝑥]𝜑𝜑))
61, 5ax-mp 7 . . . . 5 [𝐴 / 𝑥]𝜑𝜑)
7 sbcor 2807 . . . . 5 ([𝐴 / 𝑥]𝜑𝜑) ↔ ([𝐴 / 𝑥] ¬ 𝜑[𝐴 / 𝑥]𝜑))
86, 7mpbi 133 . . . 4 ([𝐴 / 𝑥] ¬ 𝜑[𝐴 / 𝑥]𝜑)
9 ralsns 3408 . . . . . 6 (𝐴 ∈ V → (∀𝑥 ∈ {𝐴} ¬ 𝜑[𝐴 / 𝑥] ¬ 𝜑))
101, 9ax-mp 7 . . . . 5 (∀𝑥 ∈ {𝐴} ¬ 𝜑[𝐴 / 𝑥] ¬ 𝜑)
11 ralsns 3408 . . . . . 6 (𝐴 ∈ V → (∀𝑥 ∈ {𝐴}𝜑[𝐴 / 𝑥]𝜑))
121, 11ax-mp 7 . . . . 5 (∀𝑥 ∈ {𝐴}𝜑[𝐴 / 𝑥]𝜑)
1310, 12orbi12i 681 . . . 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 3247 . . . 4 ({𝑥 ∈ {𝐴} ∣ 𝜑} = ∅ ↔ ∀𝑥 ∈ {𝐴} ¬ 𝜑)
16 eqcom 2042 . . . . 5 ({𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴} ↔ {𝐴} = {𝑥 ∈ {𝐴} ∣ 𝜑})
17 rabid2 2486 . . . . 5 ({𝐴} = {𝑥 ∈ {𝐴} ∣ 𝜑} ↔ ∀𝑥 ∈ {𝐴}𝜑)
1816, 17bitri 173 . . . 4 ({𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴} ↔ ∀𝑥 ∈ {𝐴}𝜑)
1915, 18orbi12i 681 . . 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 2046 . . 3 (𝑀 = {𝑥 ∈ {𝐴} ∣ 𝜑} → (𝑀 = ∅ ↔ {𝑥 ∈ {𝐴} ∣ 𝜑} = ∅))
22 eqeq1 2046 . . 3 (𝑀 = {𝑥 ∈ {𝐴} ∣ 𝜑} → (𝑀 = {𝐴} ↔ {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴}))
2321, 22orbi12d 707 . 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 629  DECID wdc 742   = wceq 1243   ∈ wcel 1393  ∀wral 2306  {crab 2310  Vcvv 2557  [wsbc 2764  ∅c0 3224  {csn 3375 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-dc 743  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rab 2315  df-v 2559  df-sbc 2765  df-dif 2920  df-nul 3225  df-sn 3381 This theorem is referenced by: (None)
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