Theorem frmin 9673

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 6311 and tz7.5 6344. (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 5595	. . . 4 ⊢ (𝐵 ⊆ 𝐴 → (𝑅 Fr 𝐴 → 𝑅 Fr 𝐵))
2		sess2 5597	. . . 4 ⊢ (𝐵 ⊆ 𝐴 → (𝑅 Se 𝐴 → 𝑅 Se 𝐵))
3	1, 2	anim12d 610	. . 3 ⊢ (𝐵 ⊆ 𝐴 → ((𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴) → (𝑅 Fr 𝐵 ∧ 𝑅 Se 𝐵)))
4		n0 4293	. . . 4 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑏 𝑏 ∈ 𝐵)
5		predeq3 6269	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑏 → Pred(𝑅, 𝐵, 𝑦) = Pred(𝑅, 𝐵, 𝑏))
6	5	eqeq1d 2738	. . . . . . . . . 10 ⊢ (𝑦 = 𝑏 → (Pred(𝑅, 𝐵, 𝑦) = ∅ ↔ Pred(𝑅, 𝐵, 𝑏) = ∅))
7	6	rspcev 3564	. . . . . . . . 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 6303	. . . . . . . . . . 11 ⊢ Pred(𝑅, 𝐵, 𝑏) = Pred((𝑅 ↾ 𝐵), 𝐵, 𝑏)
11		relres 5970	. . . . . . . . . . . . 13 ⊢ Rel (𝑅 ↾ 𝐵)
12		ssttrcl 9636	. . . . . . . . . . . . 13 ⊢ (Rel (𝑅 ↾ 𝐵) → (𝑅 ↾ 𝐵) ⊆ t++(𝑅 ↾ 𝐵))
13	11, 12	ax-mp 5	. . . . . . . . . . . 12 ⊢ (𝑅 ↾ 𝐵) ⊆ t++(𝑅 ↾ 𝐵)
14		predrelss 6301	. . . . . . . . . . . 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 3968	. . . . . . . . . 10 ⊢ Pred(𝑅, 𝐵, 𝑏) ⊆ Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏)
17		ssn0 4344	. . . . . . . . . 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 691	. . . . . . . . 9 ⊢ (Pred(𝑅, 𝐵, 𝑏) ≠ ∅ → Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) ≠ ∅)
19		predss 6273	. . . . . . . . 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 6284	. . . . . . . . . . . 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 9648	. . . . . . . . . . . . 13 ⊢ (𝑅 Se 𝐵 → t++(𝑅 ↾ 𝐵) Se 𝐵)
24		setlikespec 6289	. . . . . . . . . . . . 13 ⊢ ((𝑏 ∈ 𝐵 ∧ t++(𝑅 ↾ 𝐵) Se 𝐵) → Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) ∈ V)
25	23, 24	sylan2 594	. . . . . . . . . . . 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 3947	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) → (𝑐 ⊆ 𝐵 ↔ Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) ⊆ 𝐵))
28		neeq1 2994	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) → (𝑐 ≠ ∅ ↔ Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) ≠ ∅))
29	27, 28	anbi12d 633	. . . . . . . . . . . . . 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 6268	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) → Pred(𝑅, 𝑐, 𝑦) = Pred(𝑅, Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏), 𝑦))
31	30	eqeq1d 2738	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) → (Pred(𝑅, 𝑐, 𝑦) = ∅ ↔ Pred(𝑅, Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏), 𝑦) = ∅))
32	31	rexeqbi1dv 3306	. . . . . . . . . . . . . 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 3538	. . . . . . . . . . . 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 597	. . . . . . . . . 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 6303	. . . . . . . . . . . . . . . . 17 ⊢ Pred(𝑅, 𝐵, 𝑦) = Pred((𝑅 ↾ 𝐵), 𝐵, 𝑦)
39		predrelss 6301	. . . . . . . . . . . . . . . . . 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 3968	. . . . . . . . . . . . . . . 16 ⊢ Pred(𝑅, 𝐵, 𝑦) ⊆ Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑦)
42		inss1 4177	. . . . . . . . . . . . . . . . . . . 20 ⊢ (t++(𝑅 ↾ 𝐵) ∩ (𝐵 × 𝐵)) ⊆ t++(𝑅 ↾ 𝐵)
43		coss1 5810	. . . . . . . . . . . . . . . . . . . 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 5811	. . . . . . . . . . . . . . . . . . . 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 3931	. . . . . . . . . . . . . . . . . 18 ⊢ ((t++(𝑅 ↾ 𝐵) ∩ (𝐵 × 𝐵)) ∘ (t++(𝑅 ↾ 𝐵) ∩ (𝐵 × 𝐵))) ⊆ (t++(𝑅 ↾ 𝐵) ∘ t++(𝑅 ↾ 𝐵))
48		ttrcltr 9637	. . . . . . . . . . . . . . . . . 18 ⊢ (t++(𝑅 ↾ 𝐵) ∘ t++(𝑅 ↾ 𝐵)) ⊆ t++(𝑅 ↾ 𝐵)
49	47, 48	sstri 3931	. . . . . . . . . . . . . . . . 17 ⊢ ((t++(𝑅 ↾ 𝐵) ∩ (𝐵 × 𝐵)) ∘ (t++(𝑅 ↾ 𝐵) ∩ (𝐵 × 𝐵))) ⊆ t++(𝑅 ↾ 𝐵)
50		predtrss 6286	. . . . . . . . . . . . . . . . 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 1451	. . . . . . . . . . . . . . . 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 3933	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) ∧ 𝑏 ∈ 𝐵) → Pred(𝑅, 𝐵, 𝑦) ⊆ Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏))
53		sspred 6274	. . . . . . . . . . . . . . 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 588	. . . . . . . . . . . . . 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 2738	. . . . . . . . . . . 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 3159	. . . . . . . . . . 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 3991	. . . . . . . . . . . 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 3018	. . . . . 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 1935	. . . 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 1540   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2932  ∃wrex 3061  Vcvv 3429   ∩ cin 3888   ⊆ wss 3889  ∅c0 4273   Fr wfr 5581   Se wse 5582   × cxp 5629   ↾ cres 5633   ∘ ccom 5635  Rel wrel 5636  Predcpred 6264  t++cttrcl 9628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5375  ax-un 7689  ax-inf2 9562

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-oadd 8409  df-ttrcl 9629

This theorem is referenced by:  frind  9674  frr1  9683