bnj1417 - Mathbox for Jonathan Ben-Naim

	Mathbox for Jonathan Ben-Naim		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > bnj1417		Structured version Visualization version GIF version

Theorem bnj1417 34338

Description: Technical lemma for bnj60 34359. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
bnj1417.1	⊢ (𝜑 ↔ 𝑅 FrSe 𝐴)
bnj1417.2	⊢ (𝜓 ↔ ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))
bnj1417.3	⊢ (𝜒 ↔ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → [𝑦 / 𝑥]𝜓))
bnj1417.4	⊢ (𝜃 ↔ (𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝜒))
bnj1417.5	⊢ 𝐵 = ( pred(𝑥, 𝐴, 𝑅) ∪ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))

**Assertion**
Ref	Expression
bnj1417	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))

Distinct variable groups: 𝑥,𝐴,𝑦 𝑥,𝑅,𝑦 𝜑,𝑥,𝑦 𝜓,𝑦 Allowed substitution hints: 𝜓(𝑥) 𝜒(𝑥,𝑦) 𝜃(𝑥,𝑦) 𝐵(𝑥,𝑦)

**Proof of Theorem bnj1417**
Step	Hyp	Ref	Expression
1		bnj1417.1	. . . 4 ⊢ (𝜑 ↔ 𝑅 FrSe 𝐴)
2	1	biimpi 215	. . 3 ⊢ (𝜑 → 𝑅 FrSe 𝐴)
3		bnj1417.4	. . . . . 6 ⊢ (𝜃 ↔ (𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝜒))
4		bnj1418 34337	. . . . . . . . . . 11 ⊢ (𝑥 ∈ pred(𝑥, 𝐴, 𝑅) → 𝑥𝑅𝑥)
5	4	adantl 482	. . . . . . . . . 10 ⊢ ((𝜃 ∧ 𝑥 ∈ pred(𝑥, 𝐴, 𝑅)) → 𝑥𝑅𝑥)
6	3, 2	bnj835 34056	. . . . . . . . . . . 12 ⊢ (𝜃 → 𝑅 FrSe 𝐴)
7		df-bnj15 33990	. . . . . . . . . . . . 13 ⊢ (𝑅 FrSe 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴))
8	7	simplbi 498	. . . . . . . . . . . 12 ⊢ (𝑅 FrSe 𝐴 → 𝑅 Fr 𝐴)
9	6, 8	syl 17	. . . . . . . . . . 11 ⊢ (𝜃 → 𝑅 Fr 𝐴)
10		bnj213 34179	. . . . . . . . . . . 12 ⊢ pred(𝑥, 𝐴, 𝑅) ⊆ 𝐴
11	10	sseli 3978	. . . . . . . . . . 11 ⊢ (𝑥 ∈ pred(𝑥, 𝐴, 𝑅) → 𝑥 ∈ 𝐴)
12		frirr 5653	. . . . . . . . . . 11 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥𝑅𝑥)
13	9, 11, 12	syl2an 596	. . . . . . . . . 10 ⊢ ((𝜃 ∧ 𝑥 ∈ pred(𝑥, 𝐴, 𝑅)) → ¬ 𝑥𝑅𝑥)
14	5, 13	pm2.65da 815	. . . . . . . . 9 ⊢ (𝜃 → ¬ 𝑥 ∈ pred(𝑥, 𝐴, 𝑅))
15		nfv 1917	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦𝜑
16		nfv 1917	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴
17		bnj1417.3	. . . . . . . . . . . . . . . 16 ⊢ (𝜒 ↔ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → [𝑦 / 𝑥]𝜓))
18	17	bnj1095 34078	. . . . . . . . . . . . . . 15 ⊢ (𝜒 → ∀𝑦𝜒)
19	18	nf5i 2142	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦𝜒
20	15, 16, 19	nf3an 1904	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦(𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝜒)
21	3, 20	nfxfr 1855	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦𝜃
22	6	ad2antrr 724	. . . . . . . . . . . . . . . 16 ⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → 𝑅 FrSe 𝐴)
23		simplr 767	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → 𝑦 ∈ pred(𝑥, 𝐴, 𝑅))
24	10, 23	sselid 3980	. . . . . . . . . . . . . . . 16 ⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → 𝑦 ∈ 𝐴)
25		simpr 485	. . . . . . . . . . . . . . . 16 ⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅))
26		bnj1125 34289	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → trCl(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑦, 𝐴, 𝑅))
27	22, 24, 25, 26	syl3anc 1371	. . . . . . . . . . . . . . 15 ⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → trCl(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑦, 𝐴, 𝑅))
28		bnj1147 34291	. . . . . . . . . . . . . . . . . 18 ⊢ trCl(𝑦, 𝐴, 𝑅) ⊆ 𝐴
29	28, 25	sselid 3980	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → 𝑥 ∈ 𝐴)
30		bnj906 34227	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
31	22, 29, 30	syl2anc 584	. . . . . . . . . . . . . . . 16 ⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
32	31, 23	sseldd 3983	. . . . . . . . . . . . . . 15 ⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → 𝑦 ∈ trCl(𝑥, 𝐴, 𝑅))
33	27, 32	sseldd 3983	. . . . . . . . . . . . . 14 ⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → 𝑦 ∈ trCl(𝑦, 𝐴, 𝑅))
34	17	biimpi 215	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜒 → ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → [𝑦 / 𝑥]𝜓))
35	3, 34	bnj837 34058	. . . . . . . . . . . . . . . . 17 ⊢ (𝜃 → ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → [𝑦 / 𝑥]𝜓))
36	35	ad2antrr 724	. . . . . . . . . . . . . . . 16 ⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → [𝑦 / 𝑥]𝜓))
37		bnj1418 34337	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → 𝑦𝑅𝑥)
38	37	ad2antlr 725	. . . . . . . . . . . . . . . 16 ⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → 𝑦𝑅𝑥)
39		rsp 3244	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → [𝑦 / 𝑥]𝜓) → (𝑦 ∈ 𝐴 → (𝑦𝑅𝑥 → [𝑦 / 𝑥]𝜓)))
40	36, 24, 38, 39	syl3c 66	. . . . . . . . . . . . . . 15 ⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → [𝑦 / 𝑥]𝜓)
41		vex 3478	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ∈ V
42		bnj1417.2	. . . . . . . . . . . . . . . . 17 ⊢ (𝜓 ↔ ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))
43		eleq1w 2816	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ trCl(𝑥, 𝐴, 𝑅) ↔ 𝑦 ∈ trCl(𝑥, 𝐴, 𝑅)))
44		bnj1318 34322	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑦 → trCl(𝑥, 𝐴, 𝑅) = trCl(𝑦, 𝐴, 𝑅))
45	44	eleq2d 2819	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → (𝑦 ∈ trCl(𝑥, 𝐴, 𝑅) ↔ 𝑦 ∈ trCl(𝑦, 𝐴, 𝑅)))
46	43, 45	bitrd 278	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ trCl(𝑥, 𝐴, 𝑅) ↔ 𝑦 ∈ trCl(𝑦, 𝐴, 𝑅)))
47	46	notbid 317	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅) ↔ ¬ 𝑦 ∈ trCl(𝑦, 𝐴, 𝑅)))
48	42, 47	bitrid 282	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ ¬ 𝑦 ∈ trCl(𝑦, 𝐴, 𝑅)))
49	41, 48	sbcie 3820	. . . . . . . . . . . . . . 15 ⊢ ([𝑦 / 𝑥]𝜓 ↔ ¬ 𝑦 ∈ trCl(𝑦, 𝐴, 𝑅))
50	40, 49	sylib 217	. . . . . . . . . . . . . 14 ⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → ¬ 𝑦 ∈ trCl(𝑦, 𝐴, 𝑅))
51	33, 50	pm2.65da 815	. . . . . . . . . . . . 13 ⊢ ((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → ¬ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅))
52	51	ex 413	. . . . . . . . . . . 12 ⊢ (𝜃 → (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → ¬ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)))
53	21, 52	ralrimi 3254	. . . . . . . . . . 11 ⊢ (𝜃 → ∀𝑦 ∈ pred (𝑥, 𝐴, 𝑅) ¬ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅))
54		ralnex 3072	. . . . . . . . . . 11 ⊢ (∀𝑦 ∈ pred (𝑥, 𝐴, 𝑅) ¬ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅) ↔ ¬ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝑥 ∈ trCl(𝑦, 𝐴, 𝑅))
55	53, 54	sylib 217	. . . . . . . . . 10 ⊢ (𝜃 → ¬ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝑥 ∈ trCl(𝑦, 𝐴, 𝑅))
56		eliun 5001	. . . . . . . . . 10 ⊢ (𝑥 ∈ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅) ↔ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝑥 ∈ trCl(𝑦, 𝐴, 𝑅))
57	55, 56	sylnibr 328	. . . . . . . . 9 ⊢ (𝜃 → ¬ 𝑥 ∈ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))
58		ioran 982	. . . . . . . . 9 ⊢ (¬ (𝑥 ∈ pred(𝑥, 𝐴, 𝑅) ∨ 𝑥 ∈ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)) ↔ (¬ 𝑥 ∈ pred(𝑥, 𝐴, 𝑅) ∧ ¬ 𝑥 ∈ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)))
59	14, 57, 58	sylanbrc 583	. . . . . . . 8 ⊢ (𝜃 → ¬ (𝑥 ∈ pred(𝑥, 𝐴, 𝑅) ∨ 𝑥 ∈ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)))
60	3	simp2bi 1146	. . . . . . . . . . 11 ⊢ (𝜃 → 𝑥 ∈ 𝐴)
61		bnj1417.5	. . . . . . . . . . . 12 ⊢ 𝐵 = ( pred(𝑥, 𝐴, 𝑅) ∪ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))
62	61	bnj1414 34334	. . . . . . . . . . 11 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → trCl(𝑥, 𝐴, 𝑅) = 𝐵)
63	6, 60, 62	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜃 → trCl(𝑥, 𝐴, 𝑅) = 𝐵)
64	63	eleq2d 2819	. . . . . . . . 9 ⊢ (𝜃 → (𝑥 ∈ trCl(𝑥, 𝐴, 𝑅) ↔ 𝑥 ∈ 𝐵))
65	61	bnj1138 34085	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵 ↔ (𝑥 ∈ pred(𝑥, 𝐴, 𝑅) ∨ 𝑥 ∈ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)))
66	64, 65	bitrdi 286	. . . . . . . 8 ⊢ (𝜃 → (𝑥 ∈ trCl(𝑥, 𝐴, 𝑅) ↔ (𝑥 ∈ pred(𝑥, 𝐴, 𝑅) ∨ 𝑥 ∈ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))))
67	59, 66	mtbird 324	. . . . . . 7 ⊢ (𝜃 → ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))
68	67, 42	sylibr 233	. . . . . 6 ⊢ (𝜃 → 𝜓)
69	3, 68	sylbir 234	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝜒) → 𝜓)
70	69	3exp 1119	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜒 → 𝜓)))
71	70	ralrimiv 3145	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜒 → 𝜓))
72	17	bnj1204 34309	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜒 → 𝜓)) → ∀𝑥 ∈ 𝐴 𝜓)
73	2, 71, 72	syl2anc 584	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)
74	42	ralbii 3093	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))
75	73, 74	sylib 217	1 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∨ wo 845 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ∃wrex 3070 [wsbc 3777 ∪ cun 3946 ⊆ wss 3948 ∪ ciun 4997 class class class wbr 5148 Fr wfr 5628 predc-bnj14 33985 Se w-bnj13 33987 FrSe w-bnj15 33989 trClc-bnj18 33991

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-reg 9589 ax-inf2 9638

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-om 7858 df-1o 8468 df-bnj17 33984 df-bnj14 33986 df-bnj13 33988 df-bnj15 33990 df-bnj18 33992 df-bnj19 33994

This theorem is referenced by: bnj1421 34339

W3C validator