Theorem rabrsn 4729

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 4727	. . 3 ⊢ {𝑥 ∈ {𝐴} ∣ 𝜑} = if([𝐴 / 𝑥]𝜑, {𝐴}, ∅)
2	1	eqeq2i 2748	. 2 ⊢ (𝑀 = {𝑥 ∈ {𝐴} ∣ 𝜑} ↔ 𝑀 = if([𝐴 / 𝑥]𝜑, {𝐴}, ∅))
3		ifeqor 4582	. . . 4 ⊢ (if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) = {𝐴} ∨ if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) = ∅)
4		orcom 870	. . . 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 2739	. . . 4 ⊢ (𝑀 = if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) → (𝑀 = ∅ ↔ if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) = ∅))
7		eqeq1 2739	. . . 4 ⊢ (𝑀 = if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) → (𝑀 = {𝐴} ↔ if([𝐴 / 𝑥]𝜑, {𝐴}, ∅) = {𝐴}))
8	6, 7	orbi12d 918	. . 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 847   = wceq 1537  {crab 3433  [wsbc 3791  ∅c0 4339  ifcif 4531  {csn 4631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-rab 3434  df-sbc 3792  df-dif 3966  df-nul 4340  df-if 4532  df-sn 4632

This theorem is referenced by:  hashrabrsn  14408  hashrabsn01  14409  hashrabsn1  14410  dvnprodlem3  45904