Theorem epfrc 2933

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 2920	. 2 |- ((E Fr A /\ B (_ A /\ B =/= (/)) -> E.x e. B (B i^i {y | yEx}) = (/))
3		epel 2834	. . . . . . 7 |- (yEx <-> y e. x)
4	3	abbii 1575	. . . . . 6 |- {y | yEx} = {y | y e. x}
5		abid2 1580	. . . . . 6 |- {y | y e. x} = x
6	4, 5	eqtr2 1496	. . . . 5 |- x = {y | yEx}
7	6	ineq2i 2214	. . . 4 |- (B i^i x) = (B i^i {y | yEx})
8	7	eqeq1i 1482	. . 3 |- ((B i^i x) = (/) <-> (B i^i {y | yEx}) = (/))
9	8	rexbii 1668	. 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 775   = wceq 956   e. wcel 958  {cab 1463   =/= wne 1585  E.wrex 1646  Vcvv 1811   i^i cin 2046   (_ wss 2047  (/)c0 2280   class class class wbr 2619  Ecep 2830   Fr wfr 2915

This theorem is referenced by:  wefrc 2943  onfr 2986

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-3an 777  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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