Theorem dfepfr 2948

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 2935	. 2 |- (E Fr A <-> A.x((x (_ A /\ x =/= (/)) -> E.y e. x (x i^i {z | zEy}) = (/)))
2		epel 2850	. . . . . . . . 9 |- (zEy <-> z e. y)
3	2	abbii 1582	. . . . . . . 8 |- {z | zEy} = {z | z e. y}
4		abid2 1587	. . . . . . . 8 |- {z | z e. y} = y
5	3, 4	eqtri 1502	. . . . . . 7 |- {z | zEy} = y
6	5	ineq2i 2225	. . . . . 6 |- (x i^i {z | zEy}) = (x i^i y)
7	6	eqeq1i 1489	. . . . 5 |- ((x i^i {z | zEy}) = (/) <-> (x i^i y) = (/))
8	7	rexbii 1675	. . . 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 1003	. 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	bitri 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 958   = wceq 960   e. wcel 962  {cab 1469   =/= wne 1592  E.wrex 1653   i^i cin 2057   (_ wss 2058  (/)c0 2291   class class class wbr 2634  Ecep 2846   Fr wfr 2931

This theorem is referenced by:  onfr 3002  zfregfr 4618

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-10 970  ax-11 971  ax-12 972  ax-13 973  ax-14 974  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1127  ax-10o 1144  ax-16 1214  ax-11o 1222  ax-ext 1464  ax-sep 2718  ax-pow 2758  ax-pr 2795

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 985  df-sb 1176  df-eu 1386  df-mo 1387  df-clab 1470  df-cleq 1475  df-clel 1478  df-ne 1594  df-ral 1656  df-rex 1657  df-v 1819  df-dif 2060  df-un 2061  df-in 2062  df-ss 2064  df-nul 2292  df-pw 2414  df-sn 2424  df-pr 2425  df-op 2428  df-br 2635  df-opab 2682  df-eprel 2848  df-fr 2933

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