Theorem dfepfr 2932

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 2919	. 2 |- (E Fr A <-> A.x((x (_ A /\ x =/= (/)) -> E.y e. x (x i^i {z | zEy}) = (/)))
2		epel 2834	. . . . . . . . 9 |- (zEy <-> z e. y)
3	2	abbii 1575	. . . . . . . 8 |- {z | zEy} = {z | z e. y}
4		abid2 1580	. . . . . . . 8 |- {z | z e. y} = y
5	3, 4	eqtr 1495	. . . . . . 7 |- {z | zEy} = y
6	5	ineq2i 2214	. . . . . 6 |- (x i^i {z | zEy}) = (x i^i y)
7	6	eqeq1i 1482	. . . . 5 |- ((x i^i {z | zEy}) = (/) <-> (x i^i y) = (/))
8	7	rexbii 1668	. . . 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 999	. 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 954   = wceq 956   e. wcel 958  {cab 1463   =/= wne 1585  E.wrex 1646   i^i cin 2046   (_ wss 2047  (/)c0 2280   class class class wbr 2619  Ecep 2830   Fr wfr 2915

This theorem is referenced by:  onfr 2986  zfregfr 4601

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-eprel 2832  df-fr 2917

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