Theorem rabrsndc 3651

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 832	. . . . . . . 8 ⊢ (DECID 𝜑 → (¬ 𝜑 ∨ 𝜑))
4	2, 3	ax-mp 5	. . . . . . 7 ⊢ (¬ 𝜑 ∨ 𝜑)
5	4	sbcth 2968	. . . . . 6 ⊢ (𝐴 ∈ V → [𝐴 / 𝑥](¬ 𝜑 ∨ 𝜑))
6	1, 5	ax-mp 5	. . . . 5 ⊢ [𝐴 / 𝑥](¬ 𝜑 ∨ 𝜑)
7		sbcor 2999	. . . . 5 ⊢ ([𝐴 / 𝑥](¬ 𝜑 ∨ 𝜑) ↔ ([𝐴 / 𝑥] ¬ 𝜑 ∨ [𝐴 / 𝑥]𝜑))
8	6, 7	mpbi 144	. . . 4 ⊢ ([𝐴 / 𝑥] ¬ 𝜑 ∨ [𝐴 / 𝑥]𝜑)
9		ralsns 3621	. . . . . 6 ⊢ (𝐴 ∈ V → (∀𝑥 ∈ {𝐴} ¬ 𝜑 ↔ [𝐴 / 𝑥] ¬ 𝜑))
10	1, 9	ax-mp 5	. . . . 5 ⊢ (∀𝑥 ∈ {𝐴} ¬ 𝜑 ↔ [𝐴 / 𝑥] ¬ 𝜑)
11		ralsns 3621	. . . . . 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 759	. . . 4 ⊢ ((∀𝑥 ∈ {𝐴} ¬ 𝜑 ∨ ∀𝑥 ∈ {𝐴}𝜑) ↔ ([𝐴 / 𝑥] ¬ 𝜑 ∨ [𝐴 / 𝑥]𝜑))
14	8, 13	mpbir 145	. . 3 ⊢ (∀𝑥 ∈ {𝐴} ¬ 𝜑 ∨ ∀𝑥 ∈ {𝐴}𝜑)
15		rabeq0 3444	. . . 4 ⊢ ({𝑥 ∈ {𝐴} ∣ 𝜑} = ∅ ↔ ∀𝑥 ∈ {𝐴} ¬ 𝜑)
16		eqcom 2172	. . . . 5 ⊢ ({𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴} ↔ {𝐴} = {𝑥 ∈ {𝐴} ∣ 𝜑})
17		rabid2 2646	. . . . 5 ⊢ ({𝐴} = {𝑥 ∈ {𝐴} ∣ 𝜑} ↔ ∀𝑥 ∈ {𝐴}𝜑)
18	16, 17	bitri 183	. . . 4 ⊢ ({𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴} ↔ ∀𝑥 ∈ {𝐴}𝜑)
19	15, 18	orbi12i 759	. . 3 ⊢ (({𝑥 ∈ {𝐴} ∣ 𝜑} = ∅ ∨ {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴}) ↔ (∀𝑥 ∈ {𝐴} ¬ 𝜑 ∨ ∀𝑥 ∈ {𝐴}𝜑))
20	14, 19	mpbir 145	. 2 ⊢ ({𝑥 ∈ {𝐴} ∣ 𝜑} = ∅ ∨ {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴})
21		eqeq1 2177	. . 3 ⊢ (𝑀 = {𝑥 ∈ {𝐴} ∣ 𝜑} → (𝑀 = ∅ ↔ {𝑥 ∈ {𝐴} ∣ 𝜑} = ∅))
22		eqeq1 2177	. . 3 ⊢ (𝑀 = {𝑥 ∈ {𝐴} ∣ 𝜑} → (𝑀 = {𝐴} ↔ {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴}))
23	21, 22	orbi12d 788	. 2 ⊢ (𝑀 = {𝑥 ∈ {𝐴} ∣ 𝜑} → ((𝑀 = ∅ ∨ 𝑀 = {𝐴}) ↔ ({𝑥 ∈ {𝐴} ∣ 𝜑} = ∅ ∨ {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝐴})))
24	20, 23	mpbiri 167	1 ⊢ (𝑀 = {𝑥 ∈ {𝐴} ∣ 𝜑} → (𝑀 = ∅ ∨ 𝑀 = {𝐴}))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104   ∨ wo 703  DECID wdc 829   = wceq 1348   ∈ wcel 2141  ∀wral 2448  {crab 2452  Vcvv 2730  [wsbc 2955  ∅c0 3414  {csn 3583

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-dc 830  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-nul 3415  df-sn 3589

This theorem is referenced by: (None)