Theorem rabrsndc 3739

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 844	. . . . . . . 8 ⊢ (DECID 𝜑 → (¬ 𝜑 ∨ 𝜑))
4	2, 3	ax-mp 5	. . . . . . 7 ⊢ (¬ 𝜑 ∨ 𝜑)
5	4	sbcth 3045	. . . . . 6 ⊢ (𝐴 ∈ V → [𝐴 / 𝑥](¬ 𝜑 ∨ 𝜑))
6	1, 5	ax-mp 5	. . . . 5 ⊢ [𝐴 / 𝑥](¬ 𝜑 ∨ 𝜑)
7		sbcor 3076	. . . . 5 ⊢ ([𝐴 / 𝑥](¬ 𝜑 ∨ 𝜑) ↔ ([𝐴 / 𝑥] ¬ 𝜑 ∨ [𝐴 / 𝑥]𝜑))
8	6, 7	mpbi 145	. . . 4 ⊢ ([𝐴 / 𝑥] ¬ 𝜑 ∨ [𝐴 / 𝑥]𝜑)
9		ralsns 3707	. . . . . 6 ⊢ (𝐴 ∈ V → (∀𝑥 ∈ {𝐴} ¬ 𝜑 ↔ [𝐴 / 𝑥] ¬ 𝜑))
10	1, 9	ax-mp 5	. . . . 5 ⊢ (∀𝑥 ∈ {𝐴} ¬ 𝜑 ↔ [𝐴 / 𝑥] ¬ 𝜑)
11		ralsns 3707	. . . . . 6 ⊢ (𝐴 ∈ V → (∀𝑥 ∈ {𝐴}𝜑 ↔ [𝐴 / 𝑥]𝜑))
12	1, 11	ax-mp 5	. . . . 5 ⊢ (∀𝑥 ∈ {𝐴}𝜑 ↔ [𝐴 / 𝑥]𝜑)
13	10, 12	orbi12i 771	. . . 4 ⊢ ((∀𝑥 ∈ {𝐴} ¬ 𝜑 ∨ ∀𝑥 ∈ {𝐴}𝜑) ↔ ([𝐴 / 𝑥] ¬ 𝜑 ∨ [𝐴 / 𝑥]𝜑))
14	8, 13	mpbir 146	. . 3 ⊢ (∀𝑥 ∈ {𝐴} ¬ 𝜑 ∨ ∀𝑥 ∈ {𝐴}𝜑)
15		rabeq0 3524	. . . 4 ⊢ ({𝑥 ∈ {𝐴} ∣ 𝜑} = ∅ ↔ ∀𝑥 ∈ {𝐴} ¬ 𝜑)
16		eqcom 2233	. . . . 5 ⊢ ({𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴} ↔ {𝐴} = {𝑥 ∈ {𝐴} ∣ 𝜑})
17		rabid2 2710	. . . . 5 ⊢ ({𝐴} = {𝑥 ∈ {𝐴} ∣ 𝜑} ↔ ∀𝑥 ∈ {𝐴}𝜑)
18	16, 17	bitri 184	. . . 4 ⊢ ({𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴} ↔ ∀𝑥 ∈ {𝐴}𝜑)
19	15, 18	orbi12i 771	. . 3 ⊢ (({𝑥 ∈ {𝐴} ∣ 𝜑} = ∅ ∨ {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴}) ↔ (∀𝑥 ∈ {𝐴} ¬ 𝜑 ∨ ∀𝑥 ∈ {𝐴}𝜑))
20	14, 19	mpbir 146	. 2 ⊢ ({𝑥 ∈ {𝐴} ∣ 𝜑} = ∅ ∨ {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴})
21		eqeq1 2238	. . 3 ⊢ (𝑀 = {𝑥 ∈ {𝐴} ∣ 𝜑} → (𝑀 = ∅ ↔ {𝑥 ∈ {𝐴} ∣ 𝜑} = ∅))
22		eqeq1 2238	. . 3 ⊢ (𝑀 = {𝑥 ∈ {𝐴} ∣ 𝜑} → (𝑀 = {𝐴} ↔ {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴}))
23	21, 22	orbi12d 800	. 2 ⊢ (𝑀 = {𝑥 ∈ {𝐴} ∣ 𝜑} → ((𝑀 = ∅ ∨ 𝑀 = {𝐴}) ↔ ({𝑥 ∈ {𝐴} ∣ 𝜑} = ∅ ∨ {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴})))
24	20, 23	mpbiri 168	1 ⊢ (𝑀 = {𝑥 ∈ {𝐴} ∣ 𝜑} → (𝑀 = ∅ ∨ 𝑀 = {𝐴}))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105   ∨ wo 715  DECID wdc 841   = wceq 1397   ∈ wcel 2202  ∀wral 2510  {crab 2514  Vcvv 2802  [wsbc 3031  ∅c0 3494  {csn 3669

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-dc 842  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-nul 3495  df-sn 3675

This theorem is referenced by: (None)