Theorem rabsnifsb 3741

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 3691	. . . . . . . 8 ⊢ (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
2		sbceq1a 3042	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ [𝐴 / 𝑥]𝜑))
3	2	biimpd 144	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝜑 → [𝐴 / 𝑥]𝜑))
4	1, 3	syl 14	. . . . . . 7 ⊢ (𝑥 ∈ {𝐴} → (𝜑 → [𝐴 / 𝑥]𝜑))
5	4	imdistani 445	. . . . . 6 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝜑) → (𝑥 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑))
6	5	orcd 741	. . . . 5 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝜑) → ((𝑥 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑) ∨ (𝑥 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑)))
7	2	biimprd 158	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → ([𝐴 / 𝑥]𝜑 → 𝜑))
8	1, 7	syl 14	. . . . . . 7 ⊢ (𝑥 ∈ {𝐴} → ([𝐴 / 𝑥]𝜑 → 𝜑))
9	8	imdistani 445	. . . . . 6 ⊢ ((𝑥 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑) → (𝑥 ∈ {𝐴} ∧ 𝜑))
10		noel 3500	. . . . . . . 8 ⊢ ¬ 𝑥 ∈ ∅
11	10	pm2.21i 651	. . . . . . 7 ⊢ (𝑥 ∈ ∅ → (𝑥 ∈ {𝐴} ∧ 𝜑))
12	11	adantr 276	. . . . . 6 ⊢ ((𝑥 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑) → (𝑥 ∈ {𝐴} ∧ 𝜑))
13	9, 12	jaoi 724	. . . . 5 ⊢ (((𝑥 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑) ∨ (𝑥 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑)) → (𝑥 ∈ {𝐴} ∧ 𝜑))
14	6, 13	impbii 126	. . . 4 ⊢ ((𝑥 ∈ {𝐴} ∧ 𝜑) ↔ ((𝑥 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑) ∨ (𝑥 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑)))
15	14	abbii 2347	. . 3 ⊢ {𝑥 ∣ (𝑥 ∈ {𝐴} ∧ 𝜑)} = {𝑥 ∣ ((𝑥 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑) ∨ (𝑥 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑))}
16		nfv 1577	. . . 4 ⊢ Ⅎ𝑦((𝑥 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑) ∨ (𝑥 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑))
17		nfv 1577	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 ∈ {𝐴}
18		nfsbc1v 3051	. . . . . 6 ⊢ Ⅎ𝑥[𝐴 / 𝑥]𝜑
19	17, 18	nfan 1614	. . . . 5 ⊢ Ⅎ𝑥(𝑦 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑)
20		nfv 1577	. . . . . 6 ⊢ Ⅎ𝑥 𝑦 ∈ ∅
21	18	nfn 1706	. . . . . 6 ⊢ Ⅎ𝑥 ¬ [𝐴 / 𝑥]𝜑
22	20, 21	nfan 1614	. . . . 5 ⊢ Ⅎ𝑥(𝑦 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑)
23	19, 22	nfor 1623	. . . 4 ⊢ Ⅎ𝑥((𝑦 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑) ∨ (𝑦 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑))
24		eleq1w 2292	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ {𝐴} ↔ 𝑦 ∈ {𝐴}))
25	24	anbi1d 465	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑) ↔ (𝑦 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑)))
26		eleq1w 2292	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ ∅ ↔ 𝑦 ∈ ∅))
27	26	anbi1d 465	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑) ↔ (𝑦 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑)))
28	25, 27	orbi12d 801	. . . 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 2355	. . 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 2252	. 2 ⊢ {𝑥 ∣ (𝑥 ∈ {𝐴} ∧ 𝜑)} = {𝑦 ∣ ((𝑦 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑) ∨ (𝑦 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑))}
31		df-rab 2520	. 2 ⊢ {𝑥 ∈ {𝐴} ∣ 𝜑} = {𝑥 ∣ (𝑥 ∈ {𝐴} ∧ 𝜑)}
32		df-if 3608	. 2 ⊢ if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) = {𝑦 ∣ ((𝑦 ∈ {𝐴} ∧ [𝐴 / 𝑥]𝜑) ∨ (𝑦 ∈ ∅ ∧ ¬ [𝐴 / 𝑥]𝜑))}
33	30, 31, 32	3eqtr4i 2262	1 ⊢ {𝑥 ∈ {𝐴} ∣ 𝜑} = if([𝐴 / 𝑥]𝜑, {𝐴}, ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ∨ wo 716   = wceq 1398   ∈ wcel 2202  {cab 2217  {crab 2515  [wsbc 3032  ∅c0 3496  ifcif 3607  {csn 3673

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-nul 3497  df-if 3608  df-sn 3679

This theorem is referenced by:  rabsnif  3742