Theorem epfrc 2185

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

Hypothesis

Ref	Expression
epfrc.1	⊢ B ∈ V

Assertion

Ref	Expression
epfrc	⊢ ((E Fr A ∧ (B ⊆ A ∧ ¬ B = ∅)) → ∃x ∈ B (B ∩ x) = ∅)

Distinct variable group(s):   x,A   x,B

Proof of Theorem epfrc

Step	Hyp	Ref	Expression
1		epfrc.1	. . . 4 ⊢ B ∈ V
2	1	frc 2172	. . 3 ⊢ (E Fr A → ((B ⊆ A ∧ ¬ B = ∅) → ∃x ∈ B (B ∩ {y∣yEx}) = ∅))
3	2	imp 277	. 2 ⊢ ((E Fr A ∧ (B ⊆ A ∧ ¬ B = ∅)) → ∃x ∈ B (B ∩ {y∣yEx}) = ∅)
4		epel 2124	. . . . . . 7 ⊢ (yEx ↔ y ∈ x)
5	4	biabi 1181	. . . . . 6 ⊢ {y∣yEx} = {y∣y ∈ x}
6		abid2 1186	. . . . . 6 ⊢ {y∣y ∈ x} = x
7	5, 6	eqtr2 1120	. . . . 5 ⊢ x = {y∣yEx}
8	7	ineq2i 1642	. . . 4 ⊢ (B ∩ x) = (B ∩ {y∣yEx})
9	8	cleq1i 1108	. . 3 ⊢ ((B ∩ x) = ∅ ↔ (B ∩ {y∣yEx}) = ∅)
10	9	birex 1224	. 2 ⊢ (∃x ∈ B (B ∩ x) = ∅ ↔ ∃x ∈ B (B ∩ {y∣yEx}) = ∅)
11	3, 10	sylibr 175	1 ⊢ ((E Fr A ∧ (B ⊆ A ∧ ¬ B = ∅)) → ∃x ∈ B (B ∩ x) = ∅)

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196   ∈ wel 803  {cab 1090   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  Vcvv 1348   ∩ cin 1486   ⊆ wss 1487  ∅c0 1707   class class class wbr 2054  Ecep 2056   Fr wfr 2061

This theorem is referenced by:  wefrc 2195  onfr 2237

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-eprel 2122  df-fr 2169

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