Theorem dfepfr 2184

Description: A simpler way of saying that the epsilon relation is founded.

Assertion

Ref	Expression
dfepfr	⊢ (E Fr A ↔ ∀x((x ⊆ A ∧ ¬ x = ∅) → ∃y ∈ x (x ∩ y) = ∅))

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

Proof of Theorem dfepfr

Step	Hyp	Ref	Expression
1		dffr2 2171	. 2 ⊢ (E Fr A ↔ ∀x((x ⊆ A ∧ ¬ x = ∅) → ∃y ∈ x (x ∩ {z∣zEy}) = ∅))
2		epel 2124	. . . . . . . . 9 ⊢ (zEy ↔ z ∈ y)
3	2	biabi 1181	. . . . . . . 8 ⊢ {z∣zEy} = {z∣z ∈ y}
4		abid2 1186	. . . . . . . 8 ⊢ {z∣z ∈ y} = y
5	3, 4	eqtr 1119	. . . . . . 7 ⊢ {z∣zEy} = y
6	5	ineq2i 1642	. . . . . 6 ⊢ (x ∩ {z∣zEy}) = (x ∩ y)
7	6	cleq1i 1108	. . . . 5 ⊢ ((x ∩ {z∣zEy}) = ∅ ↔ (x ∩ y) = ∅)
8	7	birex 1224	. . . 4 ⊢ (∃y ∈ x (x ∩ {z∣zEy}) = ∅ ↔ ∃y ∈ x (x ∩ y) = ∅)
9	8	imbi2i 160	. . 3 ⊢ (((x ⊆ A ∧ ¬ x = ∅) → ∃y ∈ x (x ∩ {z∣zEy}) = ∅) ↔ ((x ⊆ A ∧ ¬ x = ∅) → ∃y ∈ x (x ∩ y) = ∅))
10	9	bial 695	. 2 ⊢ (∀x((x ⊆ A ∧ ¬ x = ∅) → ∃y ∈ x (x ∩ {z∣zEy}) = ∅) ↔ ∀x((x ⊆ A ∧ ¬ x = ∅) → ∃y ∈ x (x ∩ y) = ∅))
11	1, 10	bitr 151	1 ⊢ (E Fr A ↔ ∀x((x ⊆ A ∧ ¬ x = ∅) → ∃y ∈ x (x ∩ y) = ∅))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672   ∈ wel 803  {cab 1090   = wceq 1091  ∃wrex 1202   ∩ cin 1486   ⊆ wss 1487  ∅c0 1707   class class class wbr 2054  Ecep 2056   Fr wfr 2061

This theorem is referenced by:  onfr 2237  zfregfr 3452

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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