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Theorem rabrsndc 3465
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 756 . . . . . . . 8 (DECID 𝜑 → (¬ 𝜑𝜑))
42, 3ax-mp 7 . . . . . . 7 𝜑𝜑)
54sbcth 2799 . . . . . 6 (𝐴 ∈ V → [𝐴 / 𝑥]𝜑𝜑))
61, 5ax-mp 7 . . . . 5 [𝐴 / 𝑥]𝜑𝜑)
7 sbcor 2829 . . . . 5 ([𝐴 / 𝑥]𝜑𝜑) ↔ ([𝐴 / 𝑥] ¬ 𝜑[𝐴 / 𝑥]𝜑))
86, 7mpbi 137 . . . 4 ([𝐴 / 𝑥] ¬ 𝜑[𝐴 / 𝑥]𝜑)
9 ralsns 3435 . . . . . 6 (𝐴 ∈ V → (∀𝑥 ∈ {𝐴} ¬ 𝜑[𝐴 / 𝑥] ¬ 𝜑))
101, 9ax-mp 7 . . . . 5 (∀𝑥 ∈ {𝐴} ¬ 𝜑[𝐴 / 𝑥] ¬ 𝜑)
11 ralsns 3435 . . . . . 6 (𝐴 ∈ V → (∀𝑥 ∈ {𝐴}𝜑[𝐴 / 𝑥]𝜑))
121, 11ax-mp 7 . . . . 5 (∀𝑥 ∈ {𝐴}𝜑[𝐴 / 𝑥]𝜑)
1310, 12orbi12i 691 . . . 4 ((∀𝑥 ∈ {𝐴} ¬ 𝜑 ∨ ∀𝑥 ∈ {𝐴}𝜑) ↔ ([𝐴 / 𝑥] ¬ 𝜑[𝐴 / 𝑥]𝜑))
148, 13mpbir 138 . . 3 (∀𝑥 ∈ {𝐴} ¬ 𝜑 ∨ ∀𝑥 ∈ {𝐴}𝜑)
15 rabeq0 3274 . . . 4 ({𝑥 ∈ {𝐴} ∣ 𝜑} = ∅ ↔ ∀𝑥 ∈ {𝐴} ¬ 𝜑)
16 eqcom 2058 . . . . 5 ({𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴} ↔ {𝐴} = {𝑥 ∈ {𝐴} ∣ 𝜑})
17 rabid2 2503 . . . . 5 ({𝐴} = {𝑥 ∈ {𝐴} ∣ 𝜑} ↔ ∀𝑥 ∈ {𝐴}𝜑)
1816, 17bitri 177 . . . 4 ({𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴} ↔ ∀𝑥 ∈ {𝐴}𝜑)
1915, 18orbi12i 691 . . 3 (({𝑥 ∈ {𝐴} ∣ 𝜑} = ∅ ∨ {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴}) ↔ (∀𝑥 ∈ {𝐴} ¬ 𝜑 ∨ ∀𝑥 ∈ {𝐴}𝜑))
2014, 19mpbir 138 . 2 ({𝑥 ∈ {𝐴} ∣ 𝜑} = ∅ ∨ {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴})
21 eqeq1 2062 . . 3 (𝑀 = {𝑥 ∈ {𝐴} ∣ 𝜑} → (𝑀 = ∅ ↔ {𝑥 ∈ {𝐴} ∣ 𝜑} = ∅))
22 eqeq1 2062 . . 3 (𝑀 = {𝑥 ∈ {𝐴} ∣ 𝜑} → (𝑀 = {𝐴} ↔ {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴}))
2321, 22orbi12d 717 . 2 (𝑀 = {𝑥 ∈ {𝐴} ∣ 𝜑} → ((𝑀 = ∅ ∨ 𝑀 = {𝐴}) ↔ ({𝑥 ∈ {𝐴} ∣ 𝜑} = ∅ ∨ {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴})))
2420, 23mpbiri 161 1 (𝑀 = {𝑥 ∈ {𝐴} ∣ 𝜑} → (𝑀 = ∅ ∨ 𝑀 = {𝐴}))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 102  wo 639  DECID wdc 753   = wceq 1259  wcel 1409  wral 2323  {crab 2327  Vcvv 2574  [wsbc 2786  c0 3251  {csn 3402
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-dc 754  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rab 2332  df-v 2576  df-sbc 2787  df-dif 2947  df-nul 3252  df-sn 3408
This theorem is referenced by: (None)
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