Theorem rabsnif 3738

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 2959	. . . . . 6 |- ( y e. { x e. { A } | ph } -> y e. { A } )
2		elsni 3687	. . . . . 6 |- ( y e. { A } -> y = A )
3	1, 2	syl 14	. . . . 5 |- ( y e. { x e. { A } | ph } -> y = A )
4	3	19.8ad 1639	. . . 4 |- ( y e. { x e. { A } | ph } -> E. y y = A )
5		isset 2809	. . . 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 3498	. . . . . . . . 9 |- -. y e. (/)
8	7	intnan 936	. . . . . . . 8 |- -. ( -. ps /\ y e. (/) )
9	8	a1i 9	. . . . . . 7 |- ( y e. if ( ps , { A } , (/) ) -> -. ( -. ps /\ y e. (/) ) )
10		elif 3617	. . . . . . . 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 1385	. . . . . 6 |- ( y e. if ( ps , { A } , (/) ) -> ( ps /\ y e. { A } ) )
13	12, 2	simpl2im 386	. . . . 5 |- ( y e. if ( ps , { A } , (/) ) -> y = A )
14	13	19.8ad 1639	. . . 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 3737	. . . . 5 |- { x e. { A } | ph } = if ( [. A / x ]. ph , { A } , (/) )
17		rabsnif.f	. . . . . . 7 |- ( x = A -> ( ph <-> ps ) )
18	17	sbcieg 3064	. . . . . 6 |- ( A e. _V -> ( [. A / x ]. ph <-> ps ) )
19	18	ifbid 3627	. . . . 5 |- ( A e. _V -> if ( [. A / x ]. ph , { A } , (/) ) = if ( ps , { A } , (/) ) )
20	16, 19	eqtrid 2276	. . . 4 |- ( A e. _V -> { x e. { A } | ph } = if ( ps , { A } , (/) ) )
21	20	eleq2d 2301	. . 3 |- ( A e. _V -> ( y e. { x e. { A } | ph } <-> y e. if ( ps , { A } , (/) ) ) )
22	6, 15, 21	pm5.21nii 711	. 2 |- ( y e. { x e. { A } | ph } <-> y e. if ( ps , { A } , (/) ) )
23	22	eqriv 2228	1 |- { x e. { A } | ph } = if ( ps , { A } , (/) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 715   = wceq 1397   E.wex 1540   e. wcel 2202   {crab 2514   _Vcvv 2802   [.wsbc 3031   (/)c0 3494   ifcif 3605   {csn 3669

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-nul 3495  df-if 3606  df-sn 3675

This theorem is referenced by:  1loopgrvd2fi  16155  1hevtxdg1en  16158