Theorem dffr2 2171

Description: Alternate definition of founded relation. Similar to Definition 6.21 of [TakeutiZaring] p. 30.

Assertion

Ref	Expression
dffr2	⊢ (R Fr A ↔ ∀x((x ⊆ A ∧ ¬ x = ∅) → ∃y ∈ x (x ∩ {z∣zRy}) = ∅))

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

Proof of Theorem dffr2

Step	Hyp	Ref	Expression
1		df-fr 2169	. 2 ⊢ (R Fr A ↔ ∀x((x ⊆ A ∧ ¬ x = ∅) → ∃v ∈ x ∀w ∈ x ¬ wRv))
2		disj 1733	. . . . . . 7 ⊢ ((x ∩ {z∣zRy}) = ∅ ↔ ∀w ∈ x ¬ w ∈ {z∣zRy})
3		visset 1350	. . . . . . . . . 10 ⊢ w ∈ V
4		breq1 2065	. . . . . . . . . 10 ⊢ (z = w → (zRy ↔ wRy))
5	3, 4	elab 1415	. . . . . . . . 9 ⊢ (w ∈ {z∣zRy} ↔ wRy)
6	5	negbii 162	. . . . . . . 8 ⊢ (¬ w ∈ {z∣zRy} ↔ ¬ wRy)
7	6	biral 1223	. . . . . . 7 ⊢ (∀w ∈ x ¬ w ∈ {z∣zRy} ↔ ∀w ∈ x ¬ wRy)
8	2, 7	bitr 151	. . . . . 6 ⊢ ((x ∩ {z∣zRy}) = ∅ ↔ ∀w ∈ x ¬ wRy)
9	8	birex 1224	. . . . 5 ⊢ (∃y ∈ x (x ∩ {z∣zRy}) = ∅ ↔ ∃y ∈ x ∀w ∈ x ¬ wRy)
10		breq2 2066	. . . . . . . 8 ⊢ (y = v → (wRy ↔ wRv))
11	10	negbid 463	. . . . . . 7 ⊢ (y = v → (¬ wRy ↔ ¬ wRv))
12	11	biraldv 1219	. . . . . 6 ⊢ (y = v → (∀w ∈ x ¬ wRy ↔ ∀w ∈ x ¬ wRv))
13	12	cbvrexv 1334	. . . . 5 ⊢ (∃y ∈ x ∀w ∈ x ¬ wRy ↔ ∃v ∈ x ∀w ∈ x ¬ wRv)
14	9, 13	bitr 151	. . . 4 ⊢ (∃y ∈ x (x ∩ {z∣zRy}) = ∅ ↔ ∃v ∈ x ∀w ∈ x ¬ wRv)
15	14	imbi2i 160	. . 3 ⊢ (((x ⊆ A ∧ ¬ x = ∅) → ∃y ∈ x (x ∩ {z∣zRy}) = ∅) ↔ ((x ⊆ A ∧ ¬ x = ∅) → ∃v ∈ x ∀w ∈ x ¬ wRv))
16	15	bial 695	. 2 ⊢ (∀x((x ⊆ A ∧ ¬ x = ∅) → ∃y ∈ x (x ∩ {z∣zRy}) = ∅) ↔ ∀x((x ⊆ A ∧ ¬ x = ∅) → ∃v ∈ x ∀w ∈ x ¬ wRv))
17	1, 16	bitr4 154	1 ⊢ (R Fr A ↔ ∀x((x ⊆ A ∧ ¬ x = ∅) → ∃y ∈ x (x ∩ {z∣zRy}) = ∅))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672   = weq 797  {cab 1090   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202   ∩ cin 1486   ⊆ wss 1487  ∅c0 1707   class class class wbr 2054   Fr wfr 2061

This theorem is referenced by:  frc 2172  frss 2173  fr0 2179  dfepfr 2184  dffr3 2620

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-16 922  ax-17 925  ax-ext 1074

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-nul 1708  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-fr 2169

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