Theorem frmin 9766

Description: Every (possibly proper) subclass of a class 𝐴 with a well-founded set-like relation 𝑅 has a minimal element. This is a very strong generalization of tz6.26 6357 and tz7.5 6390. (Contributed by Scott Fenton, 4-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) (Revised by Scott Fenton, 27-Nov-2024.)

Assertion

Ref	Expression
frmin	⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑦 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅)

Distinct variable groups:   𝑦,𝐵   𝑦,𝑅

Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem frmin

Dummy variables 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		frss 5619	. . . 4 ⊢ (𝐵 ⊆ 𝐴 → (𝑅 Fr 𝐴 → 𝑅 Fr 𝐵))
2		sess2 5621	. . . 4 ⊢ (𝐵 ⊆ 𝐴 → (𝑅 Se 𝐴 → 𝑅 Se 𝐵))
3	1, 2	anim12d 621	. . 3 ⊢ (𝐵 ⊆ 𝐴 → ((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) → (𝑅 Fr 𝐵 ∧ 𝑅 Se 𝐵)))
4		n0 4303	. . . 4 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑏 𝑏 ∈ 𝐵)
5		predeq3 6315	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑏 → Pred(𝑅, 𝐵, 𝑦) = Pred(𝑅, 𝐵, 𝑏))
6	5	eqeq1d 2765	. . . . . . . . . 10 ⊢ (𝑦 = 𝑏 → (Pred(𝑅, 𝐵, 𝑦) = ∅ ↔ Pred(𝑅, 𝐵, 𝑏) = ∅))
7	6	rspcev 3580	. . . . . . . . 9 ⊢ ((𝑏 ∈ 𝐵 ∧ Pred(𝑅, 𝐵, 𝑏) = ∅) → ∃𝑦 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅)
8	7	ex 418	. . . . . . . 8 ⊢ (𝑏 ∈ 𝐵 → (Pred(𝑅, 𝐵, 𝑏) = ∅ → ∃𝑦 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅))
9	8	adantl 487	. . . . . . 7 ⊢ (((𝑅 Fr 𝐵 ∧ 𝑅 Se 𝐵) ∧ 𝑏 ∈ 𝐵) → (Pred(𝑅, 𝐵, 𝑏) = ∅ → ∃𝑦 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅))
10		predres 6349	. . . . . . . . . . 11 ⊢ Pred(𝑅, 𝐵, 𝑏) = Pred((𝑅 ↾ 𝐵), 𝐵, 𝑏)
11		relres 6001	. . . . . . . . . . . . 13 ⊢ Rel (𝑅 ↾ 𝐵)
12		ssttrcl 9729	. . . . . . . . . . . . 13 ⊢ (Rel (𝑅 ↾ 𝐵) → (𝑅 ↾ 𝐵) ⊆ t++(𝑅 ↾ 𝐵))
13	11, 12	ax-mp 5	. . . . . . . . . . . 12 ⊢ (𝑅 ↾ 𝐵) ⊆ t++(𝑅 ↾ 𝐵)
14		predrelss 6347	. . . . . . . . . . . 12 ⊢ ((𝑅 ↾ 𝐵) ⊆ t++(𝑅 ↾ 𝐵) → Pred((𝑅 ↾ 𝐵), 𝐵, 𝑏) ⊆ Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏))
15	13, 14	ax-mp 5	. . . . . . . . . . 11 ⊢ Pred((𝑅 ↾ 𝐵), 𝐵, 𝑏) ⊆ Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏)
16	10, 15	eqsstri 3959	. . . . . . . . . 10 ⊢ Pred(𝑅, 𝐵, 𝑏) ⊆ Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏)
17		ssn0 4358	. . . . . . . . . 10 ⊢ ((Pred(𝑅, 𝐵, 𝑏) ⊆ Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) ∧ Pred(𝑅, 𝐵, 𝑏) ≠ ∅) → Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) ≠ ∅)
18	16, 17	mpan 703	. . . . . . . . 9 ⊢ (Pred(𝑅, 𝐵, 𝑏) ≠ ∅ → Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) ≠ ∅)
19		predss 6319	. . . . . . . . 9 ⊢ Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) ⊆ 𝐵
20	18, 19	jctil 529	. . . . . . . 8 ⊢ (Pred(𝑅, 𝐵, 𝑏) ≠ ∅ → (Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) ⊆ 𝐵 ∧ Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) ≠ ∅))
21		dffr4 6330	. . . . . . . . . . . 12 ⊢ (𝑅 Fr 𝐵 ↔ ∀𝑐((𝑐 ⊆ 𝐵 ∧ 𝑐 ≠ ∅) → ∃𝑦 ∈ 𝑐 Pred(𝑅, 𝑐, 𝑦) = ∅))
22	21	biimpi 219	. . . . . . . . . . 11 ⊢ (𝑅 Fr 𝐵 → ∀𝑐((𝑐 ⊆ 𝐵 ∧ 𝑐 ≠ ∅) → ∃𝑦 ∈ 𝑐 Pred(𝑅, 𝑐, 𝑦) = ∅))
23		ttrclse 9741	. . . . . . . . . . . . 13 ⊢ (𝑅 Se 𝐵 → t++(𝑅 ↾ 𝐵) Se 𝐵)
24		setlikespec 6335	. . . . . . . . . . . . 13 ⊢ ((𝑏 ∈ 𝐵 ∧ t++(𝑅 ↾ 𝐵) Se 𝐵) → Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) ∈ V)
25	23, 24	sylan2 605	. . . . . . . . . . . 12 ⊢ ((𝑏 ∈ 𝐵 ∧ 𝑅 Se 𝐵) → Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) ∈ V)
26	25	ancoms 464	. . . . . . . . . . 11 ⊢ ((𝑅 Se 𝐵 ∧ 𝑏 ∈ 𝐵) → Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) ∈ V)
27		sseq1 3938	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) → (𝑐 ⊆ 𝐵 ↔ Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) ⊆ 𝐵))
28		neeq1 3021	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) → (𝑐 ≠ ∅ ↔ Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) ≠ ∅))
29	27, 28	anbi12d 644	. . . . . . . . . . . . . 14 ⊢ (𝑐 = Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) → ((𝑐 ⊆ 𝐵 ∧ 𝑐 ≠ ∅) ↔ (Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) ⊆ 𝐵 ∧ Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) ≠ ∅)))
30		predeq2 6314	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) → Pred(𝑅, 𝑐, 𝑦) = Pred(𝑅, Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏), 𝑦))
31	30	eqeq1d 2765	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) → (Pred(𝑅, 𝑐, 𝑦) = ∅ ↔ Pred(𝑅, Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏), 𝑦) = ∅))
32	31	rexeqbi1dv 3334	. . . . . . . . . . . . . 14 ⊢ (𝑐 = Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) → (∃𝑦 ∈ 𝑐 Pred(𝑅, 𝑐, 𝑦) = ∅ ↔ ∃𝑦 ∈ Pred (t++(𝑅 ↾ 𝐵), 𝐵, 𝑏)Pred(𝑅, Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏), 𝑦) = ∅))
33	29, 32	imbi12d 347	. . . . . . . . . . . . 13 ⊢ (𝑐 = Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) → (((𝑐 ⊆ 𝐵 ∧ 𝑐 ≠ ∅) → ∃𝑦 ∈ 𝑐 Pred(𝑅, 𝑐, 𝑦) = ∅) ↔ ((Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) ⊆ 𝐵 ∧ Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) ≠ ∅) → ∃𝑦 ∈ Pred (t++(𝑅 ↾ 𝐵), 𝐵, 𝑏)Pred(𝑅, Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏), 𝑦) = ∅)))
34	33	spcgv 3554	. . . . . . . . . . . 12 ⊢ (Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) ∈ V → (∀𝑐((𝑐 ⊆ 𝐵 ∧ 𝑐 ≠ ∅) → ∃𝑦 ∈ 𝑐 Pred(𝑅, 𝑐, 𝑦) = ∅) → ((Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) ⊆ 𝐵 ∧ Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) ≠ ∅) → ∃𝑦 ∈ Pred (t++(𝑅 ↾ 𝐵), 𝐵, 𝑏)Pred(𝑅, Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏), 𝑦) = ∅)))
35	34	impcom 413	. . . . . . . . . . 11 ⊢ ((∀𝑐((𝑐 ⊆ 𝐵 ∧ 𝑐 ≠ ∅) → ∃𝑦 ∈ 𝑐 Pred(𝑅, 𝑐, 𝑦) = ∅) ∧ Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) ∈ V) → ((Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) ⊆ 𝐵 ∧ Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) ≠ ∅) → ∃𝑦 ∈ Pred (t++(𝑅 ↾ 𝐵), 𝐵, 𝑏)Pred(𝑅, Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏), 𝑦) = ∅))
36	22, 26, 35	syl2an 608	. . . . . . . . . 10 ⊢ ((𝑅 Fr 𝐵 ∧ (𝑅 Se 𝐵 ∧ 𝑏 ∈ 𝐵)) → ((Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) ⊆ 𝐵 ∧ Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) ≠ ∅) → ∃𝑦 ∈ Pred (t++(𝑅 ↾ 𝐵), 𝐵, 𝑏)Pred(𝑅, Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏), 𝑦) = ∅))
37	36	anassrs 473	. . . . . . . . 9 ⊢ (((𝑅 Fr 𝐵 ∧ 𝑅 Se 𝐵) ∧ 𝑏 ∈ 𝐵) → ((Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) ⊆ 𝐵 ∧ Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) ≠ ∅) → ∃𝑦 ∈ Pred (t++(𝑅 ↾ 𝐵), 𝐵, 𝑏)Pred(𝑅, Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏), 𝑦) = ∅))
38		predres 6349	. . . . . . . . . . . . . . . . 17 ⊢ Pred(𝑅, 𝐵, 𝑦) = Pred((𝑅 ↾ 𝐵), 𝐵, 𝑦)
39		predrelss 6347	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑅 ↾ 𝐵) ⊆ t++(𝑅 ↾ 𝐵) → Pred((𝑅 ↾ 𝐵), 𝐵, 𝑦) ⊆ Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑦))
40	13, 39	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ Pred((𝑅 ↾ 𝐵), 𝐵, 𝑦) ⊆ Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑦)
41	38, 40	eqsstri 3959	. . . . . . . . . . . . . . . 16 ⊢ Pred(𝑅, 𝐵, 𝑦) ⊆ Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑦)
42		inss1 4136	. . . . . . . . . . . . . . . . . . . 20 ⊢ (t++(𝑅 ↾ 𝐵) ∩ (𝐵 × 𝐵)) ⊆ t++(𝑅 ↾ 𝐵)
43		coss1 5838	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((t++(𝑅 ↾ 𝐵) ∩ (𝐵 × 𝐵)) ⊆ t++(𝑅 ↾ 𝐵) → ((t++(𝑅 ↾ 𝐵) ∩ (𝐵 × 𝐵)) ∘ (t++(𝑅 ↾ 𝐵) ∩ (𝐵 × 𝐵))) ⊆ (t++(𝑅 ↾ 𝐵) ∘ (t++(𝑅 ↾ 𝐵) ∩ (𝐵 × 𝐵))))
44	42, 43	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ ((t++(𝑅 ↾ 𝐵) ∩ (𝐵 × 𝐵)) ∘ (t++(𝑅 ↾ 𝐵) ∩ (𝐵 × 𝐵))) ⊆ (t++(𝑅 ↾ 𝐵) ∘ (t++(𝑅 ↾ 𝐵) ∩ (𝐵 × 𝐵)))
45		coss2 5839	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((t++(𝑅 ↾ 𝐵) ∩ (𝐵 × 𝐵)) ⊆ t++(𝑅 ↾ 𝐵) → (t++(𝑅 ↾ 𝐵) ∘ (t++(𝑅 ↾ 𝐵) ∩ (𝐵 × 𝐵))) ⊆ (t++(𝑅 ↾ 𝐵) ∘ t++(𝑅 ↾ 𝐵)))
46	42, 45	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ (t++(𝑅 ↾ 𝐵) ∘ (t++(𝑅 ↾ 𝐵) ∩ (𝐵 × 𝐵))) ⊆ (t++(𝑅 ↾ 𝐵) ∘ t++(𝑅 ↾ 𝐵))
47	44, 46	sstri 3922	. . . . . . . . . . . . . . . . . 18 ⊢ ((t++(𝑅 ↾ 𝐵) ∩ (𝐵 × 𝐵)) ∘ (t++(𝑅 ↾ 𝐵) ∩ (𝐵 × 𝐵))) ⊆ (t++(𝑅 ↾ 𝐵) ∘ t++(𝑅 ↾ 𝐵))
48		ttrcltr 9730	. . . . . . . . . . . . . . . . . 18 ⊢ (t++(𝑅 ↾ 𝐵) ∘ t++(𝑅 ↾ 𝐵)) ⊆ t++(𝑅 ↾ 𝐵)
49	47, 48	sstri 3922	. . . . . . . . . . . . . . . . 17 ⊢ ((t++(𝑅 ↾ 𝐵) ∩ (𝐵 × 𝐵)) ∘ (t++(𝑅 ↾ 𝐵) ∩ (𝐵 × 𝐵))) ⊆ t++(𝑅 ↾ 𝐵)
50		predtrss 6332	. . . . . . . . . . . . . . . . 17 ⊢ ((((t++(𝑅 ↾ 𝐵) ∩ (𝐵 × 𝐵)) ∘ (t++(𝑅 ↾ 𝐵) ∩ (𝐵 × 𝐵))) ⊆ t++(𝑅 ↾ 𝐵) ∧ 𝑦 ∈ Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) ∧ 𝑏 ∈ 𝐵) → Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑦) ⊆ Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏))
51	49, 50	mp3an1 1477	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦 ∈ Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) ∧ 𝑏 ∈ 𝐵) → Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑦) ⊆ Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏))
52	41, 51	sstrid 3924	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) ∧ 𝑏 ∈ 𝐵) → Pred(𝑅, 𝐵, 𝑦) ⊆ Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏))
53		sspred 6320	. . . . . . . . . . . . . . 15 ⊢ ((Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) ⊆ 𝐵 ∧ Pred(𝑅, 𝐵, 𝑦) ⊆ Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏)) → Pred(𝑅, 𝐵, 𝑦) = Pred(𝑅, Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏), 𝑦))
54	19, 52, 53	sylancr 599	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) ∧ 𝑏 ∈ 𝐵) → Pred(𝑅, 𝐵, 𝑦) = Pred(𝑅, Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏), 𝑦))
55	54	ancoms 464	. . . . . . . . . . . . 13 ⊢ ((𝑏 ∈ 𝐵 ∧ 𝑦 ∈ Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏)) → Pred(𝑅, 𝐵, 𝑦) = Pred(𝑅, Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏), 𝑦))
56	55	eqeq1d 2765	. . . . . . . . . . . 12 ⊢ ((𝑏 ∈ 𝐵 ∧ 𝑦 ∈ Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏)) → (Pred(𝑅, 𝐵, 𝑦) = ∅ ↔ Pred(𝑅, Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏), 𝑦) = ∅))
57	56	rexbidva 3188	. . . . . . . . . . 11 ⊢ (𝑏 ∈ 𝐵 → (∃𝑦 ∈ Pred (t++(𝑅 ↾ 𝐵), 𝐵, 𝑏)Pred(𝑅, 𝐵, 𝑦) = ∅ ↔ ∃𝑦 ∈ Pred (t++(𝑅 ↾ 𝐵), 𝐵, 𝑏)Pred(𝑅, Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏), 𝑦) = ∅))
58		ssrexv 3983	. . . . . . . . . . . 12 ⊢ (Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) ⊆ 𝐵 → (∃𝑦 ∈ Pred (t++(𝑅 ↾ 𝐵), 𝐵, 𝑏)Pred(𝑅, 𝐵, 𝑦) = ∅ → ∃𝑦 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅))
59	19, 58	ax-mp 5	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ Pred (t++(𝑅 ↾ 𝐵), 𝐵, 𝑏)Pred(𝑅, 𝐵, 𝑦) = ∅ → ∃𝑦 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅)
60	57, 59	biimtrrdi 257	. . . . . . . . . 10 ⊢ (𝑏 ∈ 𝐵 → (∃𝑦 ∈ Pred (t++(𝑅 ↾ 𝐵), 𝐵, 𝑏)Pred(𝑅, Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏), 𝑦) = ∅ → ∃𝑦 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅))
61	60	adantl 487	. . . . . . . . 9 ⊢ (((𝑅 Fr 𝐵 ∧ 𝑅 Se 𝐵) ∧ 𝑏 ∈ 𝐵) → (∃𝑦 ∈ Pred (t++(𝑅 ↾ 𝐵), 𝐵, 𝑏)Pred(𝑅, Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏), 𝑦) = ∅ → ∃𝑦 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅))
62	37, 61	syld 48	. . . . . . . 8 ⊢ (((𝑅 Fr 𝐵 ∧ 𝑅 Se 𝐵) ∧ 𝑏 ∈ 𝐵) → ((Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) ⊆ 𝐵 ∧ Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) ≠ ∅) → ∃𝑦 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅))
63	20, 62	syl5 35	. . . . . . 7 ⊢ (((𝑅 Fr 𝐵 ∧ 𝑅 Se 𝐵) ∧ 𝑏 ∈ 𝐵) → (Pred(𝑅, 𝐵, 𝑏) ≠ ∅ → ∃𝑦 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅))
64	9, 63	pm2.61dne 3045	. . . . . 6 ⊢ (((𝑅 Fr 𝐵 ∧ 𝑅 Se 𝐵) ∧ 𝑏 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅)
65	64	ex 418	. . . . 5 ⊢ ((𝑅 Fr 𝐵 ∧ 𝑅 Se 𝐵) → (𝑏 ∈ 𝐵 → ∃𝑦 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅))
66	65	exlimdv 1967	. . . 4 ⊢ ((𝑅 Fr 𝐵 ∧ 𝑅 Se 𝐵) → (∃𝑏 𝑏 ∈ 𝐵 → ∃𝑦 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅))
67	4, 66	biimtrid 245	. . 3 ⊢ ((𝑅 Fr 𝐵 ∧ 𝑅 Se 𝐵) → (𝐵 ≠ ∅ → ∃𝑦 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅))
68	3, 67	syl6com 38	. 2 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) → (𝐵 ⊆ 𝐴 → (𝐵 ≠ ∅ → ∃𝑦 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅)))
69	68	imp32 424	1 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑦 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1567   = wceq 1569  ∃wex 1811   ∈ wcel 2146   ≠ wne 2959  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900   ∩ cin 4092  ∅c0 4281   Fr wfr 5605   Se wse 5606   × cxp 5653   ↾ cres 5657   ∘ ccom 5659  Rel wrel 5660  Predcpred 6310  t++cttrcl 9721

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7761  ax-inf2 9655

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6311  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6501  df-fun 6549  df-fn 6550  df-f 6551  df-f1 6552  df-fo 6553  df-f1o 6554  df-fv 6555  df-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7888  df-2nd 8012  df-frecs 8313  df-wrecs 8344  df-recs 8393  df-rdg 8432  df-1o 8490  df-oadd 8494  df-ttrcl 9722

This theorem is used by:  frind  9767  frr1  9776