Theorem epfrc 3642

Description: A subset of an epsilon-founded class has a minimal element. (A distinct variable restriction was removed by David Abernethy, 22-Feb-2011.)

Hypothesis

Ref	Expression
epfrc.1	|- B e. _V

Assertion

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

Distinct variable group:   x,B

Proof of Theorem epfrc

Step	Hyp	Ref	Expression
1		epfrc.1	. . 3 |- B e. _V
2	1	frc 3629	. 2 |- (( _E Fr A /\ B C_ A /\ B =/= (/)) -> E.x e. B (B i^i {y | y _E x}) = (/))
3		epel 3585	. . . . . . 7 |- (y _E x <-> y e. x)
4	3	abbii 2006	. . . . . 6 |- {y | y _E x} = {y | y e. x}
5		abid2 2011	. . . . . 6 |- {y | y e. x} = x
6	4, 5	eqtr2i 1909	. . . . 5 |- x = {y | y _E x}
7	6	ineq2i 2793	. . . 4 |- (B i^i x) = (B i^i {y | y _E x})
8	7	eqeq1i 1891	. . 3 |- ((B i^i x) = (/) <-> (B i^i {y | y _E x}) = (/))
9	8	rexbii 2128	. 2 |- (E.x e. B (B i^i x) = (/) <-> E.x e. B (B i^i {y | y _E x}) = (/))
10	2, 9	sylibr 217	1 |- (( _E Fr A /\ B C_ A /\ B =/= (/)) -> E.x e. B (B i^i x) = (/))

Syntax hints:   -> wi 3   /\ w3a 858   = wceq 1298   e. wcel 1300  {cab 1871   =/= wne 2017  E.wrex 2106  _Vcvv 2292   i^i cin 2592   C_ wss 2593  (/)c0 2875   class class class wbr 3338   _E cep 3581   Fr wfr 3623

This theorem is referenced by:  wefrc 3652  onfr 3702

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-3an 860  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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