Theorem rabrsn 4749

Description: A restricted class abstraction restricted to a singleton is either the empty set or the singleton itself. (Contributed by Alexander van der Vekens, 22-Dec-2017.) (Proof shortened by AV, 21-Jul-2019.)

Assertion

Ref	Expression
rabrsn	⊢ (𝑀 = {𝑥 ∈ {𝐴} ∣ 𝜑} → (𝑀 = ∅ ∨ 𝑀 = {𝐴}))

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝑀(𝑥)

Proof of Theorem rabrsn

Step	Hyp	Ref	Expression
1		rabsnifsb 4747	. . 3 ⊢ {𝑥 ∈ {𝐴} ∣ 𝜑} = if([𝐴 / 𝑥]𝜑, {𝐴}, ∅)
2	1	eqeq2i 2753	. 2 ⊢ (𝑀 = {𝑥 ∈ {𝐴} ∣ 𝜑} ↔ 𝑀 = if([𝐴 / 𝑥]𝜑, {𝐴}, ∅))
3		ifeqor 4599	. . . 4 ⊢ (if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) = {𝐴} ∨ if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) = ∅)
4		orcom 869	. . . 4 ⊢ ((if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) = {𝐴} ∨ if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) = ∅) ↔ (if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) = ∅ ∨ if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) = {𝐴}))
5	3, 4	mpbi 230	. . 3 ⊢ (if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) = ∅ ∨ if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) = {𝐴})
6		eqeq1 2744	. . . 4 ⊢ (𝑀 = if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) → (𝑀 = ∅ ↔ if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) = ∅))
7		eqeq1 2744	. . . 4 ⊢ (𝑀 = if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) → (𝑀 = {𝐴} ↔ if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) = {𝐴}))
8	6, 7	orbi12d 917	. . 3 ⊢ (𝑀 = if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) → ((𝑀 = ∅ ∨ 𝑀 = {𝐴}) ↔ (if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) = ∅ ∨ if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) = {𝐴})))
9	5, 8	mpbiri 258	. 2 ⊢ (𝑀 = if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) → (𝑀 = ∅ ∨ 𝑀 = {𝐴}))
10	2, 9	sylbi 217	1 ⊢ (𝑀 = {𝑥 ∈ {𝐴} ∣ 𝜑} → (𝑀 = ∅ ∨ 𝑀 = {𝐴}))

Syntax hints:   → wi 4   ∨ wo 846   = wceq 1537  {crab 3443  [wsbc 3804  ∅c0 4352  ifcif 4548  {csn 4648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-rab 3444  df-sbc 3805  df-dif 3979  df-nul 4353  df-if 4549  df-sn 4649

This theorem is referenced by:  hashrabrsn  14421  hashrabsn01  14422  hashrabsn1  14423  dvnprodlem3  45869