Theorem epfrc 2939

Description: A subset of an epsilon-founded class has a minimal element.

Hypothesis

Ref	Expression
epfrc.1	|- B e. V

Assertion

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

Distinct variable groups:   x,A   x,B

Proof of Theorem epfrc

Step	Hyp	Ref	Expression
1		epfrc.1	. . 3 |- B e. V
2	1	frc 2926	. 2 |- ((E Fr A /\ B (_ A /\ B =/= (/)) -> E.x e. B (B i^i {y | yEx}) = (/))
3		epel 2840	. . . . . . 7 |- (yEx <-> y e. x)
4	3	abbii 1578	. . . . . 6 |- {y | yEx} = {y | y e. x}
5		abid2 1583	. . . . . 6 |- {y | y e. x} = x
6	4, 5	eqtr2 1499	. . . . 5 |- x = {y | yEx}
7	6	ineq2i 2217	. . . 4 |- (B i^i x) = (B i^i {y | yEx})
8	7	eqeq1i 1485	. . 3 |- ((B i^i x) = (/) <-> (B i^i {y | yEx}) = (/))
9	8	rexbii 1671	. 2 |- (E.x e. B (B i^i x) = (/) <-> E.x e. B (B i^i {y | yEx}) = (/))
10	2, 9	sylibr 200	1 |- ((E Fr A /\ B (_ A /\ B =/= (/)) -> E.x e. B (B i^i x) = (/))

Syntax hints:   -> wi 3   /\ w3a 777   = wceq 958   e. wcel 960  {cab 1466   =/= wne 1588  E.wrex 1649  Vcvv 1814   i^i cin 2049   (_ wss 2050  (/)c0 2283   class class class wbr 2624  Ecep 2836   Fr wfr 2921

This theorem is referenced by:  wefrc 2949  onfr 2992

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-3an 779  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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