Theorem frmin 9728

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 6352 and tz7.5 6385. (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 5627	. . . 4 ⊢ (𝐵 ⊆ 𝐴 → (𝑅 Fr 𝐴 → 𝑅 Fr 𝐵))
2		sess2 5629	. . . 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 4307	. . . 4 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑏 𝑏 ∈ 𝐵)
5		predeq3 6310	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑏 → Pred(𝑅, 𝐵, 𝑦) = Pred(𝑅, 𝐵, 𝑏))
6	5	eqeq1d 2767	. . . . . . . . . 10 ⊢ (𝑦 = 𝑏 → (Pred(𝑅, 𝐵, 𝑦) = ∅ ↔ Pred(𝑅, 𝐵, 𝑏) = ∅))
7	6	rspcev 3583	. . . . . . . . 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 6344	. . . . . . . . . . 11 ⊢ Pred(𝑅, 𝐵, 𝑏) = Pred((𝑅 ↾ 𝐵), 𝐵, 𝑏)
11		relres 6006	. . . . . . . . . . . . 13 ⊢ Rel (𝑅 ↾ 𝐵)
12		ssttrcl 9691	. . . . . . . . . . . . 13 ⊢ (Rel (𝑅 ↾ 𝐵) → (𝑅 ↾ 𝐵) ⊆ t++(𝑅 ↾ 𝐵))
13	11, 12	ax-mp 5	. . . . . . . . . . . 12 ⊢ (𝑅 ↾ 𝐵) ⊆ t++(𝑅 ↾ 𝐵)
14		predrelss 6342	. . . . . . . . . . . 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 3984	. . . . . . . . . 10 ⊢ Pred(𝑅, 𝐵, 𝑏) ⊆ Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏)
17		ssn0 4362	. . . . . . . . . 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 6314	. . . . . . . . 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 6325	. . . . . . . . . . . 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 9703	. . . . . . . . . . . . 13 ⊢ (𝑅 Se 𝐵 → t++(𝑅 ↾ 𝐵) Se 𝐵)
24		setlikespec 6330	. . . . . . . . . . . . 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 3963	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) → (𝑐 ⊆ 𝐵 ↔ Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) ⊆ 𝐵))
28		neeq1 3022	. . . . . . . . . . . . . . 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 6309	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) → Pred(𝑅, 𝑐, 𝑦) = Pred(𝑅, Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏), 𝑦))
31	30	eqeq1d 2767	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) → (Pred(𝑅, 𝑐, 𝑦) = ∅ ↔ Pred(𝑅, Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏), 𝑦) = ∅))
32	31	rexeqbi1dv 3336	. . . . . . . . . . . . . 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 3557	. . . . . . . . . . . 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 6344	. . . . . . . . . . . . . . . . 17 ⊢ Pred(𝑅, 𝐵, 𝑦) = Pred((𝑅 ↾ 𝐵), 𝐵, 𝑦)
39		predrelss 6342	. . . . . . . . . . . . . . . . . 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 3984	. . . . . . . . . . . . . . . 16 ⊢ Pred(𝑅, 𝐵, 𝑦) ⊆ Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑦)
42		inss1 4189	. . . . . . . . . . . . . . . . . . . 20 ⊢ (t++(𝑅 ↾ 𝐵) ∩ (𝐵 × 𝐵)) ⊆ t++(𝑅 ↾ 𝐵)
43		coss1 5843	. . . . . . . . . . . . . . . . . . . 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 5844	. . . . . . . . . . . . . . . . . . . 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 3947	. . . . . . . . . . . . . . . . . 18 ⊢ ((t++(𝑅 ↾ 𝐵) ∩ (𝐵 × 𝐵)) ∘ (t++(𝑅 ↾ 𝐵) ∩ (𝐵 × 𝐵))) ⊆ (t++(𝑅 ↾ 𝐵) ∘ t++(𝑅 ↾ 𝐵))
48		ttrcltr 9692	. . . . . . . . . . . . . . . . . 18 ⊢ (t++(𝑅 ↾ 𝐵) ∘ t++(𝑅 ↾ 𝐵)) ⊆ t++(𝑅 ↾ 𝐵)
49	47, 48	sstri 3947	. . . . . . . . . . . . . . . . 17 ⊢ ((t++(𝑅 ↾ 𝐵) ∩ (𝐵 × 𝐵)) ∘ (t++(𝑅 ↾ 𝐵) ∩ (𝐵 × 𝐵))) ⊆ t++(𝑅 ↾ 𝐵)
50		predtrss 6327	. . . . . . . . . . . . . . . . 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 3949	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏) ∧ 𝑏 ∈ 𝐵) → Pred(𝑅, 𝐵, 𝑦) ⊆ Pred(t++(𝑅 ↾ 𝐵), 𝐵, 𝑏))
53		sspred 6315	. . . . . . . . . . . . . . 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 2767	. . . . . . . . . . . 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 3189	. . . . . . . . . . 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 4008	. . . . . . . . . . . 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 3046	. . . . . 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 1966	. . . 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 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2146   ≠ wne 2960  ∃wrex 3091  Vcvv 3457   ∩ cin 3905   ⊆ wss 3906  ∅c0 4286   Fr wfr 5613   Se wse 5614   × cxp 5661   ↾ cres 5665   ∘ ccom 5667  Rel wrel 5668  Predcpred 6305  t++cttrcl 9683

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7742  ax-inf2 9617

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-oadd 8463  df-ttrcl 9684

This theorem is used by:  frind  9729  frr1  9738