Theorem rabsnt 2884

Description: Truth implied by equality of a restricted class abstraction and a singleton.

Hypotheses

Ref	Expression
rabsnt.1	|- B e. V
rabsnt.2	|- (x = B -> (ph <-> ps))

Assertion

Ref	Expression
rabsnt	|- ({x e. A | ph} = {B} -> ps)

Distinct variable groups:   x,A   x,B   ps,x

Proof of Theorem rabsnt

Step	Hyp	Ref	Expression
1		rabsnt.2	. . . 4 |- (x = B -> (ph <-> ps))
2	1	reuuni2 2874	. . 3 |- ((B e. A /\ E!x e. A ph) -> (ps <-> U.{x e. A | ph} = B))
3	2	biimprd 154	. 2 |- ((B e. A /\ E!x e. A ph) -> (U.{x e. A | ph} = B -> ps))
4		ssrab2 2121	. . . . 5 |- {x e. A | ph} (_ A
5		sseq1 2072	. . . . 5 |- ({x e. A | ph} = {B} -> ({x e. A | ph} (_ A <-> {B} (_ A))
6	4, 5	mpbii 193	. . . 4 |- ({x e. A | ph} = {B} -> {B} (_ A)
7		rabsnt.1	. . . . 5 |- B e. V
8	7	snss 2452	. . . 4 |- (B e. A <-> {B} (_ A)
9	6, 8	sylibr 200	. . 3 |- ({x e. A | ph} = {B} -> B e. A)
10	7	reusni 2883	. . 3 |- ({x e. A | ph} = {B} -> E!x e. A ph)
11	9, 10	jca 288	. 2 |- ({x e. A | ph} = {B} -> (B e. A /\ E!x e. A ph))
12		unieq 2500	. . 3 |- ({x e. A | ph} = {B} -> U.{x e. A | ph} = U.{B})
13	7	unisn 2507	. . 3 |- U.{B} = B
14	12, 13	syl6eq 1515	. 2 |- ({x e. A | ph} = {B} -> U.{x e. A | ph} = B)
15	3, 11, 14	sylc 68	1 |- ({x e. A | ph} = {B} -> ps)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  E!wreu 1639  {crab 1640  Vcvv 1802   (_ wss 2037  {csn 2399  U.cuni 2493

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-reu 1643  df-rab 1644  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-uni 2494

Copyright terms: Public domain