Theorem rabsnif 3774

Description: A restricted class abstraction restricted to a singleton is either the empty set or the singleton itself. (Contributed by AV, 12-Apr-2019.) (Proof shortened by AV, 21-Jul-2019.)

Hypothesis

Ref	Expression
rabsnif.f	|- ( x = A -> ( ph <-> ps ) )

Assertion

Ref	Expression
rabsnif	|- { x e. { A } | ph } = if ( ps , { A } , (/) )

Distinct variable groups:   x, A   ps, x

Allowed substitution hint:   ph( x)

Proof of Theorem rabsnif

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elrabi 2979	. . . . . 6 |- ( y e. { x e. { A } | ph } -> y e. { A } )
2		elsni 3723	. . . . . 6 |- ( y e. { A } -> y = A )
3	1, 2	syl 14	. . . . 5 |- ( y e. { x e. { A } | ph } -> y = A )
4	3	19.8ad 1644	. . . 4 |- ( y e. { x e. { A } | ph } -> E. y y = A )
5		isset 2828	. . . 4 |- ( A e. _V <-> E. y y = A )
6	4, 5	sylibr 134	. . 3 |- ( y e. { x e. { A } | ph } -> A e. _V )
7		noel 3525	. . . . . . . . 9 |- -. y e. (/)
8	7	intnan 941	. . . . . . . 8 |- -. ( -. ps /\ y e. (/) )
9	8	a1i 9	. . . . . . 7 |- ( y e. if ( ps , { A } , (/) ) -> -. ( -. ps /\ y e. (/) ) )
10		elif 3649	. . . . . . . 8 |- ( y e. if ( ps , { A } , (/) ) <-> ( ( ps /\ y e. { A } ) \/ ( -. ps /\ y e. (/) ) ) )
11	10	biimpi 120	. . . . . . 7 |- ( y e. if ( ps , { A } , (/) ) -> ( ( ps /\ y e. { A } ) \/ ( -. ps /\ y e. (/) ) ) )
12	9, 11	ecased 1390	. . . . . 6 |- ( y e. if ( ps , { A } , (/) ) -> ( ps /\ y e. { A } ) )
13	12, 2	simpl2im 390	. . . . 5 |- ( y e. if ( ps , { A } , (/) ) -> y = A )
14	13	19.8ad 1644	. . . 4 |- ( y e. if ( ps , { A } , (/) ) -> E. y y = A )
15	14, 5	sylibr 134	. . 3 |- ( y e. if ( ps , { A } , (/) ) -> A e. _V )
16		rabsnifsb 3773	. . . . 5 |- { x e. { A } | ph } = if ( [. A / x ]. ph , { A } , (/) )
17		rabsnif.f	. . . . . . 7 |- ( x = A -> ( ph <-> ps ) )
18	17	sbcieg 3084	. . . . . 6 |- ( A e. _V -> ( [. A / x ]. ph <-> ps ) )
19	18	ifbid 3659	. . . . 5 |- ( A e. _V -> if ( [. A / x ]. ph , { A } , (/) ) = if ( ps , { A } , (/) ) )
20	16, 19	eqtrid 2283	. . . 4 |- ( A e. _V -> { x e. { A } | ph } = if ( ps , { A } , (/) ) )
21	20	eleq2d 2308	. . 3 |- ( A e. _V -> ( y e. { x e. { A } | ph } <-> y e. if ( ps , { A } , (/) ) ) )
22	6, 15, 21	pm5.21nii 716	. 2 |- ( y e. { x e. { A } | ph } <-> y e. if ( ps , { A } , (/) ) )
23	22	eqriv 2235	1 |- { x e. { A } | ph } = if ( ps , { A } , (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720   = wceq 1402   E.wex 1545   e. wcel 2209   {crab 2532   _Vcvv 2821   [.wsbc 3051   (/)c0 3520   ifcif 3635   {csn 3705

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-nul 3521  df-if 3636  df-sn 3711

This theorem is referenced by:  suppsnopdc  6480  1loopgrvd2fi  16460  1hevtxdg1en  16463