Theorem frmin 9789

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 6368 and tz7.5 6405. (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 5649	. . . 4 ⊢ (𝐵 ⊆ 𝐴 → (𝑅 Fr 𝐴 → 𝑅 Fr 𝐵))
2		sess2 5651	. . . 4 ⊢ (𝐵 ⊆ 𝐴 → (𝑅 Se 𝐴 → 𝑅 Se 𝐵))
3	1, 2	anim12d 609	. . 3 ⊢ (𝐵 ⊆ 𝐴 → ((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) → (𝑅 Fr 𝐵 ∧ 𝑅 Se 𝐵)))
4		n0 4353	. . . 4 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑏 𝑏 ∈ 𝐵)
5		predeq3 6325	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑏 → Pred(𝑅, 𝐵, 𝑦) = Pred(𝑅, 𝐵, 𝑏))
6	5	eqeq1d 2739	. . . . . . . . . 10 ⊢ (𝑦 = 𝑏 → (Pred(𝑅, 𝐵, 𝑦) = ∅ ↔ Pred(𝑅, 𝐵, 𝑏) = ∅))
7	6	rspcev 3622	. . . . . . . . 9 ⊢ ((𝑏 ∈ 𝐵 ∧ Pred(𝑅, 𝐵, 𝑏) = ∅) → ∃𝑦 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅)
8	7	ex 412	. . . . . . . 8 ⊢ (𝑏 ∈ 𝐵 → (Pred(𝑅, 𝐵, 𝑏) = ∅ → ∃𝑦 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅))
9	8	adantl 481	. . . . . . 7 ⊢ (((𝑅 Fr 𝐵 ∧ 𝑅 Se 𝐵) ∧ 𝑏 ∈ 𝐵) → (Pred(𝑅, 𝐵, 𝑏) = ∅ → ∃𝑦 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅))
10		predres 6360	. . . . . . . . . . 11 ⊢ Pred(𝑅, 𝐵, 𝑏) = Pred((𝑅 ↾ 𝐵), 𝐵, 𝑏)
11		relres 6023	. . . . . . . . . . . . 13 ⊢ Rel (𝑅 ↾ 𝐵)
12		ssttrcl 9755	. . . . . . . . . . . . 13 ⊢ (Rel (𝑅 ↾ 𝐵) → (𝑅 ↾ 𝐵) ⊆ t++(𝑅 ↾ 𝐵))
13	11, 12	ax-mp 5	. . . . . . . . . . . 12 ⊢ (𝑅 ↾ 𝐵) ⊆ t++(𝑅 ↾ 𝐵)
14		predrelss 6358	. . . . . . . . . . . 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 4030	. . . . . . . . . 10 ⊢ Pred(𝑅, 𝐵, 𝑏) ⊆ Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏)
17		ssn0 4404	. . . . . . . . . 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 690	. . . . . . . . 9 ⊢ (Pred(𝑅, 𝐵, 𝑏) ≠ ∅ → Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) ≠ ∅)
19		predss 6329	. . . . . . . . 9 ⊢ Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) ⊆ 𝐵
20	18, 19	jctil 519	. . . . . . . 8 ⊢ (Pred(𝑅, 𝐵, 𝑏) ≠ ∅ → (Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) ⊆ 𝐵 ∧ Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) ≠ ∅))
21		dffr4 6341	. . . . . . . . . . . 12 ⊢ (𝑅 Fr 𝐵 ↔ ∀𝑐((𝑐 ⊆ 𝐵 ∧ 𝑐 ≠ ∅) → ∃𝑦 ∈ 𝑐 Pred(𝑅, 𝑐, 𝑦) = ∅))
22	21	biimpi 216	. . . . . . . . . . 11 ⊢ (𝑅 Fr 𝐵 → ∀𝑐((𝑐 ⊆ 𝐵 ∧ 𝑐 ≠ ∅) → ∃𝑦 ∈ 𝑐 Pred(𝑅, 𝑐, 𝑦) = ∅))
23		ttrclse 9767	. . . . . . . . . . . . 13 ⊢ (𝑅 Se 𝐵 → t++(𝑅 ↾ 𝐵) Se 𝐵)
24		setlikespec 6346	. . . . . . . . . . . . 13 ⊢ ((𝑏 ∈ 𝐵 ∧ t++(𝑅 ↾ 𝐵) Se 𝐵) → Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) ∈ V)
25	23, 24	sylan2 593	. . . . . . . . . . . 12 ⊢ ((𝑏 ∈ 𝐵 ∧ 𝑅 Se 𝐵) → Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) ∈ V)
26	25	ancoms 458	. . . . . . . . . . 11 ⊢ ((𝑅 Se 𝐵 ∧ 𝑏 ∈ 𝐵) → Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) ∈ V)
27		sseq1 4009	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) → (𝑐 ⊆ 𝐵 ↔ Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) ⊆ 𝐵))
28		neeq1 3003	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) → (𝑐 ≠ ∅ ↔ Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) ≠ ∅))
29	27, 28	anbi12d 632	. . . . . . . . . . . . . 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 6324	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) → Pred(𝑅, 𝑐, 𝑦) = Pred(𝑅, Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏), 𝑦))
31	30	eqeq1d 2739	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) → (Pred(𝑅, 𝑐, 𝑦) = ∅ ↔ Pred(𝑅, Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏), 𝑦) = ∅))
32	31	rexeqbi1dv 3339	. . . . . . . . . . . . . 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 344	. . . . . . . . . . . . 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 3596	. . . . . . . . . . . 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 407	. . . . . . . . . . 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 596	. . . . . . . . . 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 467	. . . . . . . . 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 6360	. . . . . . . . . . . . . . . . 17 ⊢ Pred(𝑅, 𝐵, 𝑦) = Pred((𝑅 ↾ 𝐵), 𝐵, 𝑦)
39		predrelss 6358	. . . . . . . . . . . . . . . . . 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 4030	. . . . . . . . . . . . . . . 16 ⊢ Pred(𝑅, 𝐵, 𝑦) ⊆ Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑦)
42		inss1 4237	. . . . . . . . . . . . . . . . . . . 20 ⊢ (t++(𝑅 ↾ 𝐵) ∩ (𝐵 × 𝐵)) ⊆ t++(𝑅 ↾ 𝐵)
43		coss1 5866	. . . . . . . . . . . . . . . . . . . 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 5867	. . . . . . . . . . . . . . . . . . . 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 3993	. . . . . . . . . . . . . . . . . 18 ⊢ ((t++(𝑅 ↾ 𝐵) ∩ (𝐵 × 𝐵)) ∘ (t++(𝑅 ↾ 𝐵) ∩ (𝐵 × 𝐵))) ⊆ (t++(𝑅 ↾ 𝐵) ∘ t++(𝑅 ↾ 𝐵))
48		ttrcltr 9756	. . . . . . . . . . . . . . . . . 18 ⊢ (t++(𝑅 ↾ 𝐵) ∘ t++(𝑅 ↾ 𝐵)) ⊆ t++(𝑅 ↾ 𝐵)
49	47, 48	sstri 3993	. . . . . . . . . . . . . . . . 17 ⊢ ((t++(𝑅 ↾ 𝐵) ∩ (𝐵 × 𝐵)) ∘ (t++(𝑅 ↾ 𝐵) ∩ (𝐵 × 𝐵))) ⊆ t++(𝑅 ↾ 𝐵)
50		predtrss 6343	. . . . . . . . . . . . . . . . 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 1450	. . . . . . . . . . . . . . . 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 3995	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) ∧ 𝑏 ∈ 𝐵) → Pred(𝑅, 𝐵, 𝑦) ⊆ Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏))
53		sspred 6330	. . . . . . . . . . . . . . 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 587	. . . . . . . . . . . . . 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 458	. . . . . . . . . . . . 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 2739	. . . . . . . . . . . 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 3177	. . . . . . . . . . 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 4053	. . . . . . . . . . . 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 254	. . . . . . . . . 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 481	. . . . . . . . 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 47	. . . . . . . 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 34	. . . . . . 7 ⊢ (((𝑅 Fr 𝐵 ∧ 𝑅 Se 𝐵) ∧ 𝑏 ∈ 𝐵) → (Pred(𝑅, 𝐵, 𝑏) ≠ ∅ → ∃𝑦 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅))
64	9, 63	pm2.61dne 3028	. . . . . 6 ⊢ (((𝑅 Fr 𝐵 ∧ 𝑅 Se 𝐵) ∧ 𝑏 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅)
65	64	ex 412	. . . . 5 ⊢ ((𝑅 Fr 𝐵 ∧ 𝑅 Se 𝐵) → (𝑏 ∈ 𝐵 → ∃𝑦 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅))
66	65	exlimdv 1933	. . . 4 ⊢ ((𝑅 Fr 𝐵 ∧ 𝑅 Se 𝐵) → (∃𝑏 𝑏 ∈ 𝐵 → ∃𝑦 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅))
67	4, 66	biimtrid 242	. . 3 ⊢ ((𝑅 Fr 𝐵 ∧ 𝑅 Se 𝐵) → (𝐵 ≠ ∅ → ∃𝑦 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅))
68	3, 67	syl6com 37	. 2 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) → (𝐵 ⊆ 𝐴 → (𝐵 ≠ ∅ → ∃𝑦 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅)))
69	68	imp32 418	1 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐵 ⊆ 𝐴 ∧ 𝐵 ≠ ∅)) → ∃𝑦 ∈ 𝐵 Pred(𝑅, 𝐵, 𝑦) = ∅)

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2108   ≠ wne 2940  ∃wrex 3070  Vcvv 3480   ∩ cin 3950   ⊆ wss 3951  ∅c0 4333   Fr wfr 5634   Se wse 5635   × cxp 5683   ↾ cres 5687   ∘ ccom 5689  Rel wrel 5690  Predcpred 6320  t++cttrcl 9747

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755  ax-inf2 9681

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510  df-ttrcl 9748

This theorem is referenced by:  frind  9790  frr1  9799