Theorem rabrsn 4259

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 e. { A } | ph } -> ( M = (/) \/ M = { A } ) )

Distinct variable group:   x, A

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

Proof of Theorem rabrsn

Step	Hyp	Ref	Expression
1		rabsnifsb 4257	. . 3 |- { x e. { A } | ph } = if ( [. A / x ]. ph , { A } , (/) )
2	1	eqeq2i 2634	. 2 |- ( M = { x e. { A } | ph } <-> M = if ( [. A / x ]. ph , { A } , (/) ) )
3		ifeqor 4132	. . . 4 |- ( if ( [. A / x ]. ph , { A } , (/) ) = { A } \/ if ( [. A / x ]. ph , { A } , (/) ) = (/) )
4		orcom 402	. . . 4 |- ( ( if ( [. A / x ]. ph , { A } , (/) ) = { A } \/ if ( [. A / x ]. ph , { A } , (/) ) = (/) ) <-> ( if ( [. A / x ]. ph , { A } , (/) ) = (/) \/ if ( [. A / x ]. ph , { A } , (/) ) = { A } ) )
5	3, 4	mpbi 220	. . 3 |- ( if ( [. A / x ]. ph , { A } , (/) ) = (/) \/ if ( [. A / x ]. ph , { A } , (/) ) = { A } )
6		eqeq1 2626	. . . 4 |- ( M = if ( [. A / x ]. ph , { A } , (/) ) -> ( M = (/) <-> if ( [. A / x ]. ph , { A } , (/) ) = (/) ) )
7		eqeq1 2626	. . . 4 |- ( M = if ( [. A / x ]. ph , { A } , (/) ) -> ( M = { A } <-> if ( [. A / x ]. ph , { A } , (/) ) = { A } ) )
8	6, 7	orbi12d 746	. . 3 |- ( M = if ( [. A / x ]. ph , { A } , (/) ) -> ( ( M = (/) \/ M = { A } ) <-> ( if ( [. A / x ]. ph , { A } , (/) ) = (/) \/ if ( [. A / x ]. ph , { A } , (/) ) = { A } ) ) )
9	5, 8	mpbiri 248	. 2 |- ( M = if ( [. A / x ]. ph , { A } , (/) ) -> ( M = (/) \/ M = { A } ) )
10	2, 9	sylbi 207	1 |- ( M = { x e. { A } | ph } -> ( M = (/) \/ M = { A } ) )

Syntax hints:   -> wi 4   \/ wo 383   = wceq 1483   {crab 2916   [.wsbc 3435   (/)c0 3915   ifcif 4086   {csn 4177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-nul 3916  df-if 4087  df-sn 4178

This theorem is referenced by:  hashrabrsn  13161  hashrabsn01  13162  hashrabsn1  13163  dvnprodlem3  40163