Theorem dfepfrOLD 3641

Description: An alternate way of saying that the epsilon relation is founded.

Assertion

Ref	Expression
dfepfrOLD	|- ( _E Fr A <-> A.x((x C_ A /\ x =/= (/)) -> E.y e. x (x i^i y) = (/)))

Distinct variable groups:   x,y   x,A

Proof of Theorem dfepfrOLD

Step	Hyp	Ref	Expression
1		dffr2 3627	. 2 |- ( _E Fr A <-> A.x((x C_ A /\ x =/= (/)) -> E.y e. x (x i^i {z | z _E y}) = (/)))
2		epel 3585	. . . . . . . . 9 |- (z _E y <-> z e. y)
3	2	abbii 2006	. . . . . . . 8 |- {z | z _E y} = {z | z e. y}
4		abid2 2011	. . . . . . . 8 |- {z | z e. y} = y
5	3, 4	eqtri 1908	. . . . . . 7 |- {z | z _E y} = y
6	5	ineq2i 2793	. . . . . 6 |- (x i^i {z | z _E y}) = (x i^i y)
7	6	eqeq1i 1891	. . . . 5 |- ((x i^i {z | z _E y}) = (/) <-> (x i^i y) = (/))
8	7	rexbii 2128	. . . 4 |- (E.y e. x (x i^i {z | z _E y}) = (/) <-> E.y e. x (x i^i y) = (/))
9	8	imbi2i 202	. . 3 |- (((x C_ A /\ x =/= (/)) -> E.y e. x (x i^i {z | z _E y}) = (/)) <-> ((x C_ A /\ x =/= (/)) -> E.y e. x (x i^i y) = (/)))
10	9	albii 1346	. 2 |- (A.x((x C_ A /\ x =/= (/)) -> E.y e. x (x i^i {z | z _E y}) = (/)) <-> A.x((x C_ A /\ x =/= (/)) -> E.y e. x (x i^i y) = (/)))
11	1, 10	bitri 190	1 |- ( _E Fr A <-> A.x((x C_ A /\ x =/= (/)) -> E.y e. x (x i^i y) = (/)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  {cab 1871   =/= wne 2017  E.wrex 2106   i^i cin 2592   C_ wss 2593  (/)c0 2875   class class class wbr 3338   _E cep 3581   Fr wfr 3623

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-eprel 3583  df-fr 3625

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