Theorem dfepfr 2938

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

Assertion

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

Distinct variable groups:   x,y   x,A

Proof of Theorem dfepfr

Step	Hyp	Ref	Expression
1		dffr2 2925	. 2 |- (E Fr A <-> A.x((x (_ A /\ x =/= (/)) -> E.y e. x (x i^i {z | zEy}) = (/)))
2		epel 2840	. . . . . . . . 9 |- (zEy <-> z e. y)
3	2	abbii 1578	. . . . . . . 8 |- {z | zEy} = {z | z e. y}
4		abid2 1583	. . . . . . . 8 |- {z | z e. y} = y
5	3, 4	eqtr 1498	. . . . . . 7 |- {z | zEy} = y
6	5	ineq2i 2217	. . . . . 6 |- (x i^i {z | zEy}) = (x i^i y)
7	6	eqeq1i 1485	. . . . 5 |- ((x i^i {z | zEy}) = (/) <-> (x i^i y) = (/))
8	7	rexbii 1671	. . . 4 |- (E.y e. x (x i^i {z | zEy}) = (/) <-> E.y e. x (x i^i y) = (/))
9	8	imbi2i 185	. . 3 |- (((x (_ A /\ x =/= (/)) -> E.y e. x (x i^i {z | zEy}) = (/)) <-> ((x (_ A /\ x =/= (/)) -> E.y e. x (x i^i y) = (/)))
10	9	albii 1001	. 2 |- (A.x((x (_ A /\ x =/= (/)) -> E.y e. x (x i^i {z | zEy}) = (/)) <-> A.x((x (_ A /\ x =/= (/)) -> E.y e. x (x i^i y) = (/)))
11	1, 10	bitr 173	1 |- (E Fr A <-> A.x((x (_ A /\ x =/= (/)) -> E.y e. x (x i^i y) = (/)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 956   = wceq 958   e. wcel 960  {cab 1466   =/= wne 1588  E.wrex 1649   i^i cin 2049   (_ wss 2050  (/)c0 2283   class class class wbr 2624  Ecep 2836   Fr wfr 2921

This theorem is referenced by:  onfr 2992  zfregfr 4610

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-eprel 2838  df-fr 2923

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