Theorem rabrsndc 3412

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 736	. . . . . . . 8 ⊢ (DECID φ → (¬ φ ∨ φ))
4	2, 3	ax-mp 7	. . . . . . 7 ⊢ (¬ φ ∨ φ)
5	4	sbcth 2754	. . . . . 6 ⊢ (A ∈ V → [A / x](¬ φ ∨ φ))
6	1, 5	ax-mp 7	. . . . 5 ⊢ [A / x](¬ φ ∨ φ)
7		sbcor 2784	. . . . 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 3382	. . . . . 6 ⊢ (A ∈ V → (∀x ∈ {A} ¬ φ ↔ [A / x] ¬ φ))
10	1, 9	ax-mp 7	. . . . 5 ⊢ (∀x ∈ {A} ¬ φ ↔ [A / x] ¬ φ)
11		ralsns 3382	. . . . . 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 668	. . . 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 3224	. . . 4 ⊢ ({x ∈ {A} ∣ φ} = ∅ ↔ ∀x ∈ {A} ¬ φ)
16		eqcom 2024	. . . . 5 ⊢ ({x ∈ {A} ∣ φ} = {A} ↔ {A} = {x ∈ {A} ∣ φ})
17		rabid2 2464	. . . . 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 668	. . 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 2028	. . 3 ⊢ (𝑀 = {x ∈ {A} ∣ φ} → (𝑀 = ∅ ↔ {x ∈ {A} ∣ φ} = ∅))
22		eqeq1 2028	. . 3 ⊢ (𝑀 = {x ∈ {A} ∣ φ} → (𝑀 = {A} ↔ {x ∈ {A} ∣ φ} = {A}))
23	21, 22	orbi12d 694	. 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 616  DECID wdc 733   = wceq 1228   ∈ wcel 1374  ∀wral 2284  {crab 2288  Vcvv 2535  [wsbc 2741  ∅c0 3201  {csn 3350

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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004

This theorem depends on definitions:  df-bi 110  df-dc 734  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rab 2293  df-v 2537  df-sbc 2742  df-dif 2897  df-nul 3202  df-sn 3356

This theorem is referenced by: (None)