Theorem rabsnifsb 4656

Description: A restricted class abstraction restricted to a singleton is either the empty set or the singleton itself. (Contributed by AV, 21-Jul-2019.)

Assertion

Ref	Expression
rabsnifsb	⊢ {𝑥 ∈ {𝐴} ∣ 𝜑} = if([𝐴 / 𝑥]𝜑, {𝐴}, ∅)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rabsnifsb

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elsni 4574	. . . . . . . 8 ⊢ (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
2		sbceq1a 3735	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ [𝐴 / 𝑥]𝜑))
3	2	biimpd 231	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝜑 → [𝐴 / 𝑥]𝜑))
4	1, 3	syl 17	. . . . . . 7 ⊢ (𝑥 ∈ {𝐴} → (𝜑 → [𝐴 / 𝑥]𝜑))
5	4	imdistani 574	. . . . . 6 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝜑) → (𝑥 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑))
6	5	orcd 880	. . . . 5 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝜑) → ((𝑥 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑) ∨ (𝑥 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑)))
7	2	biimprd 250	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → ([𝐴 / 𝑥]𝜑 → 𝜑))
8	1, 7	syl 17	. . . . . . 7 ⊢ (𝑥 ∈ {𝐴} → ([𝐴 / 𝑥]𝜑 → 𝜑))
9	8	imdistani 574	. . . . . 6 ⊢ ((𝑥 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑) → (𝑥 ∈ {𝐴} ∧ 𝜑))
10		noel 4268	. . . . . . . 8 ⊢ ¬ 𝑥 ∈ ∅
11	10	pm2.21i 119	. . . . . . 7 ⊢ (𝑥 ∈ ∅ → (𝑥 ∈ {𝐴} ∧ 𝜑))
12	11	adantr 482	. . . . . 6 ⊢ ((𝑥 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑) → (𝑥 ∈ {𝐴} ∧ 𝜑))
13	9, 12	jaoi 864	. . . . 5 ⊢ (((𝑥 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑) ∨ (𝑥 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑)) → (𝑥 ∈ {𝐴} ∧ 𝜑))
14	6, 13	impbii 211	. . . 4 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝜑) ↔ ((𝑥 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑) ∨ (𝑥 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑)))
15	14	abbii 2808	. . 3 ⊢ {𝑥 ∣ (𝑥 ∈ {𝐴} ∧ 𝜑)} = {𝑥 ∣ ((𝑥 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑) ∨ (𝑥 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑))}
16		nfv 1922	. . . 4 ⊢ Ⅎ𝑦((𝑥 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑) ∨ (𝑥 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑))
17		nfv 1922	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 ∈ {𝐴}
18		nfsbc1v 3744	. . . . . 6 ⊢ Ⅎ𝑥[𝐴 / 𝑥]𝜑
19	17, 18	nfan 1907	. . . . 5 ⊢ Ⅎ𝑥(𝑦 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑)
20		nfv 1922	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 ∈ ∅
21	18	nfn 1865	. . . . . 6 ⊢ Ⅎ𝑥 ¬ [𝐴 / 𝑥]𝜑
22	20, 21	nfan 1907	. . . . 5 ⊢ Ⅎ𝑥(𝑦 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑)
23	19, 22	nfor 1912	. . . 4 ⊢ Ⅎ𝑥((𝑦 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑) ∨ (𝑦 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑))
24		eleq1w 2824	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ {𝐴} ↔ 𝑦 ∈ {𝐴}))
25	24	anbi1d 638	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑) ↔ (𝑦 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑)))
26		eleq1w 2824	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ ∅ ↔ 𝑦 ∈ ∅))
27	26	anbi1d 638	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑) ↔ (𝑦 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑)))
28	25, 27	orbi12d 925	. . . 4 ⊢ (𝑥 = 𝑦 → (((𝑥 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑) ∨ (𝑥 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑)) ↔ ((𝑦 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑) ∨ (𝑦 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑))))
29	16, 23, 28	cbvabw 2812	. . 3 ⊢ {𝑥 ∣ ((𝑥 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑) ∨ (𝑥 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑))} = {𝑦 ∣ ((𝑦 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑) ∨ (𝑦 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑))}
30	15, 29	eqtri 2764	. 2 ⊢ {𝑥 ∣ (𝑥 ∈ {𝐴} ∧ 𝜑)} = {𝑦 ∣ ((𝑦 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑) ∨ (𝑦 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑))}
31		df-rab 3394	. 2 ⊢ {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ {𝐴} ∧ 𝜑)}
32		df-if 4457	. 2 ⊢ if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) = {𝑦 ∣ ((𝑦 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑) ∨ (𝑦 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑))}
33	30, 31, 32	3eqtr4i 2774	1 ⊢ {𝑥 ∈ {𝐴} ∣ 𝜑} = if([𝐴 / 𝑥]𝜑, {𝐴}, ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 397   ∨ wo 854   = wceq 1548   ∈ wcel 2121  {cab 2719  {crab 3393  [wsbc 3724  ∅c0 4263  ifcif 4456  {csn 4557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-rab 3394  df-sbc 3725  df-dif 3887  df-nul 4264  df-if 4457  df-sn 4558

This theorem is referenced by:  rabsnif  4657  rabrsn  4658