Theorem fri 2170

Description: Property of founded relation (one direction of definition).

Hypothesis

Ref	Expression
fri.1	⊢ B ∈ V

Assertion

Ref	Expression
fri	⊢ ((R Fr A ∧ (B ⊆ A ∧ ¬ B = ∅)) → ∃x ∈ B ∀y ∈ B ¬ yRx)

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

Proof of Theorem fri

Step	Hyp	Ref	Expression
1		df-fr 2169	. . 3 ⊢ (R Fr A ↔ ∀z((z ⊆ A ∧ ¬ z = ∅) → ∃x ∈ z ∀y ∈ z ¬ yRx))
2		fri.1	. . . 4 ⊢ B ∈ V
3		sseq1 1521	. . . . . 6 ⊢ (z = B → (z ⊆ A ↔ B ⊆ A))
4		cleq1 1107	. . . . . . 7 ⊢ (z = B → (z = ∅ ↔ B = ∅))
5	4	negbid 463	. . . . . 6 ⊢ (z = B → (¬ z = ∅ ↔ ¬ B = ∅))
6	3, 5	anbi12d 476	. . . . 5 ⊢ (z = B → ((z ⊆ A ∧ ¬ z = ∅) ↔ (B ⊆ A ∧ ¬ B = ∅)))
7		raleq 1324	. . . . . 6 ⊢ (z = B → (∀y ∈ z ¬ yRx ↔ ∀y ∈ B ¬ yRx))
8	7	rexeqd 1328	. . . . 5 ⊢ (z = B → (∃x ∈ z ∀y ∈ z ¬ yRx ↔ ∃x ∈ B ∀y ∈ B ¬ yRx))
9	6, 8	imbi12d 474	. . . 4 ⊢ (z = B → (((z ⊆ A ∧ ¬ z = ∅) → ∃x ∈ z ∀y ∈ z ¬ yRx) ↔ ((B ⊆ A ∧ ¬ B = ∅) → ∃x ∈ B ∀y ∈ B ¬ yRx)))
10	2, 9	cla4v 1400	. . 3 ⊢ (∀z((z ⊆ A ∧ ¬ z = ∅) → ∃x ∈ z ∀y ∈ z ¬ yRx) → ((B ⊆ A ∧ ¬ B = ∅) → ∃x ∈ B ∀y ∈ B ¬ yRx))
11	1, 10	sylbi 174	. 2 ⊢ (R Fr A → ((B ⊆ A ∧ ¬ B = ∅) → ∃x ∈ B ∀y ∈ B ¬ yRx))
12	11	imp 277	1 ⊢ ((R Fr A ∧ (B ⊆ A ∧ ¬ B = ∅)) → ∃x ∈ B ∀y ∈ B ¬ yRx)

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196  ∀wal 672   = wceq 1091   ∈ wcel 1092  ∀wral 1201  ∃wrex 1202  Vcvv 1348   ⊆ wss 1487  ∅c0 1707   class class class wbr 2054   Fr wfr 2061

This theorem is referenced by:  wereu 2197

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-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-in 1491  df-ss 1492  df-fr 2169

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