Theorem rabsssn 4215

Description: Conditions for a restricted class abstraction to be a subset of a singleton, i.e. to be a singleton or the empty set. (Contributed by AV, 18-Apr-2019.)

Assertion

Ref	Expression
rabsssn	|- ( { x e. V | ph } C_ { X } <-> A. x e. V ( ph -> x = X ) )

Distinct variable group:   x, X

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

Proof of Theorem rabsssn

Step	Hyp	Ref	Expression
1		df-rab 2921	. . 3 |- { x e. V | ph } = { x | ( x e. V /\ ph ) }
2		df-sn 4178	. . 3 |- { X } = { x | x = X }
3	1, 2	sseq12i 3631	. 2 |- ( { x e. V | ph } C_ { X } <-> { x | ( x e. V /\ ph ) } C_ { x | x = X } )
4		ss2ab 3670	. 2 |- ( { x | ( x e. V /\ ph ) } C_ { x | x = X } <-> A. x ( ( x e. V /\ ph ) -> x = X ) )
5		impexp 462	. . . 4 |- ( ( ( x e. V /\ ph ) -> x = X ) <-> ( x e. V -> ( ph -> x = X ) ) )
6	5	albii 1747	. . 3 |- ( A. x ( ( x e. V /\ ph ) -> x = X ) <-> A. x ( x e. V -> ( ph -> x = X ) ) )
7		df-ral 2917	. . 3 |- ( A. x e. V ( ph -> x = X ) <-> A. x ( x e. V -> ( ph -> x = X ) ) )
8	6, 7	bitr4i 267	. 2 |- ( A. x ( ( x e. V /\ ph ) -> x = X ) <-> A. x e. V ( ph -> x = X ) )
9	3, 4, 8	3bitri 286	1 |- ( { x e. V | ph } C_ { X } <-> A. x e. V ( ph -> x = X ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   {cab 2608   A.wral 2912   {crab 2916   C_ wss 3574   {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-ral 2917  df-rab 2921  df-in 3581  df-ss 3588  df-sn 4178

This theorem is referenced by:  suppmptcfin  42160  linc1  42214