Theorem rabrsndc 3429

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

Step	Hyp	Ref	Expression
1		rabrsndc.1	. . . . . 6 ⊢ A ∈ V
2		rabrsndc.2	. . . . . . . 8 ⊢ DECID φ
3		pm2.1dc 744	. . . . . . . 8 ⊢ (DECID φ → (¬ φ ∨ φ))
4	2, 3	ax-mp 7	. . . . . . 7 ⊢ (¬ φ ∨ φ)
5	4	sbcth 2771	. . . . . 6 ⊢ (A ∈ V → [A / x](¬ φ ∨ φ))
6	1, 5	ax-mp 7	. . . . 5 ⊢ [A / x](¬ φ ∨ φ)
7		sbcor 2801	. . . . 5 ⊢ ([A / x](¬ φ ∨ φ) ↔ ([A / x] ¬ φ ∨ [A / x]φ))
8	6, 7	mpbi 133	. . . 4 ⊢ ([A / x] ¬ φ ∨ [A / x]φ)
9		ralsns 3399	. . . . . 6 ⊢ (A ∈ V → (∀x ∈ {A} ¬ φ ↔ [A / x] ¬ φ))
10	1, 9	ax-mp 7	. . . . 5 ⊢ (∀x ∈ {A} ¬ φ ↔ [A / x] ¬ φ)
11		ralsns 3399	. . . . . 6 ⊢ (A ∈ V → (∀x ∈ {A}φ ↔ [A / x]φ))
12	1, 11	ax-mp 7	. . . . 5 ⊢ (∀x ∈ {A}φ ↔ [A / x]φ)
13	10, 12	orbi12i 680	. . . 4 ⊢ ((∀x ∈ {A} ¬ φ ∨ ∀x ∈ {A}φ) ↔ ([A / x] ¬ φ ∨ [A / x]φ))
14	8, 13	mpbir 134	. . 3 ⊢ (∀x ∈ {A} ¬ φ ∨ ∀x ∈ {A}φ)
15		rabeq0 3241	. . . 4 ⊢ ({x ∈ {A} ∣ φ} = ∅ ↔ ∀x ∈ {A} ¬ φ)
16		eqcom 2039	. . . . 5 ⊢ ({x ∈ {A} ∣ φ} = {A} ↔ {A} = {x ∈ {A} ∣ φ})
17		rabid2 2480	. . . . 5 ⊢ ({A} = {x ∈ {A} ∣ φ} ↔ ∀x ∈ {A}φ)
18	16, 17	bitri 173	. . . 4 ⊢ ({x ∈ {A} ∣ φ} = {A} ↔ ∀x ∈ {A}φ)
19	15, 18	orbi12i 680	. . 3 ⊢ (({x ∈ {A} ∣ φ} = ∅ ∨ {x ∈ {A} ∣ φ} = {A}) ↔ (∀x ∈ {A} ¬ φ ∨ ∀x ∈ {A}φ))
20	14, 19	mpbir 134	. 2 ⊢ ({x ∈ {A} ∣ φ} = ∅ ∨ {x ∈ {A} ∣ φ} = {A})
21		eqeq1 2043	. . 3 ⊢ (𝑀 = {x ∈ {A} ∣ φ} → (𝑀 = ∅ ↔ {x ∈ {A} ∣ φ} = ∅))
22		eqeq1 2043	. . 3 ⊢ (𝑀 = {x ∈ {A} ∣ φ} → (𝑀 = {A} ↔ {x ∈ {A} ∣ φ} = {A}))
23	21, 22	orbi12d 706	. 2 ⊢ (𝑀 = {x ∈ {A} ∣ φ} → ((𝑀 = ∅ ∨ 𝑀 = {A}) ↔ ({x ∈ {A} ∣ φ} = ∅ ∨ {x ∈ {A} ∣ φ} = {A})))
24	20, 23	mpbiri 157	1 ⊢ (𝑀 = {x ∈ {A} ∣ φ} → (𝑀 = ∅ ∨ 𝑀 = {A}))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 98   ∨ wo 628  DECID wdc 741   = wceq 1242   ∈ wcel 1390  ∀wral 2300  {crab 2304  Vcvv 2551  [wsbc 2758  ∅c0 3218  {csn 3367

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-dc 742  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rab 2309  df-v 2553  df-sbc 2759  df-dif 2914  df-nul 3219  df-sn 3373

This theorem is referenced by: (None)