bnj1417 - Mathbox for Jonathan Ben-Naim

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Theorem bnj1417 35223

Description: Technical lemma for bnj60 35244. 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 216	. . 3 ⊢ (𝜑 → 𝑅 FrSe 𝐴)
3		bnj1417.4	. . . . . 6 ⊢ (𝜃 ↔ (𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝜒))
4		bnj1418 35222	. . . . . . . . . . 11 ⊢ (𝑥 ∈ pred(𝑥, 𝐴, 𝑅) → 𝑥𝑅𝑥)
5	4	adantl 481	. . . . . . . . . 10 ⊢ ((𝜃 ∧ 𝑥 ∈ pred(𝑥, 𝐴, 𝑅)) → 𝑥𝑅𝑥)
6	3, 2	bnj835 34942	. . . . . . . . . . . 12 ⊢ (𝜃 → 𝑅 FrSe 𝐴)
7		df-bnj15 34876	. . . . . . . . . . . . 13 ⊢ (𝑅 FrSe 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Se 𝐴))
8	7	simplbi 496	. . . . . . . . . . . 12 ⊢ (𝑅 FrSe 𝐴 → 𝑅 Fr 𝐴)
9	6, 8	syl 17	. . . . . . . . . . 11 ⊢ (𝜃 → 𝑅 Fr 𝐴)
10		bnj213 35064	. . . . . . . . . . . 12 ⊢ pred(𝑥, 𝐴, 𝑅) ⊆ 𝐴
11	10	sseli 3931	. . . . . . . . . . 11 ⊢ (𝑥 ∈ pred(𝑥, 𝐴, 𝑅) → 𝑥 ∈ 𝐴)
12		frirr 5610	. . . . . . . . . . 11 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥𝑅𝑥)
13	9, 11, 12	syl2an 597	. . . . . . . . . 10 ⊢ ((𝜃 ∧ 𝑥 ∈ pred(𝑥, 𝐴, 𝑅)) → ¬ 𝑥𝑅𝑥)
14	5, 13	pm2.65da 817	. . . . . . . . 9 ⊢ (𝜃 → ¬ 𝑥 ∈ pred(𝑥, 𝐴, 𝑅))
15		nfv 1916	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦𝜑
16		nfv 1916	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴
17		bnj1417.3	. . . . . . . . . . . . . . . 16 ⊢ (𝜒 ↔ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → [𝑦 / 𝑥]𝜓))
18	17	bnj1095 34964	. . . . . . . . . . . . . . 15 ⊢ (𝜒 → ∀𝑦𝜒)
19	18	nf5i 2152	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦𝜒
20	15, 16, 19	nf3an 1903	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦(𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝜒)
21	3, 20	nfxfr 1855	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦𝜃
22	6	ad2antrr 727	. . . . . . . . . . . . . . . 16 ⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → 𝑅 FrSe 𝐴)
23		simplr 769	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → 𝑦 ∈ pred(𝑥, 𝐴, 𝑅))
24	10, 23	sselid 3933	. . . . . . . . . . . . . . . 16 ⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → 𝑦 ∈ 𝐴)
25		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅))
26		bnj1125 35174	. . . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . . 15 ⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → trCl(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑦, 𝐴, 𝑅))
28		bnj1147 35176	. . . . . . . . . . . . . . . . . 18 ⊢ trCl(𝑦, 𝐴, 𝑅) ⊆ 𝐴
29	28, 25	sselid 3933	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → 𝑥 ∈ 𝐴)
30		bnj906 35112	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
31	22, 29, 30	syl2anc 585	. . . . . . . . . . . . . . . 16 ⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
32	31, 23	sseldd 3936	. . . . . . . . . . . . . . 15 ⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → 𝑦 ∈ trCl(𝑥, 𝐴, 𝑅))
33	27, 32	sseldd 3936	. . . . . . . . . . . . . 14 ⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → 𝑦 ∈ trCl(𝑦, 𝐴, 𝑅))
34	17	biimpi 216	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜒 → ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → [𝑦 / 𝑥]𝜓))
35	3, 34	bnj837 34944	. . . . . . . . . . . . . . . . 17 ⊢ (𝜃 → ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → [𝑦 / 𝑥]𝜓))
36	35	ad2antrr 727	. . . . . . . . . . . . . . . 16 ⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → [𝑦 / 𝑥]𝜓))
37		bnj1418 35222	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → 𝑦𝑅𝑥)
38	37	ad2antlr 728	. . . . . . . . . . . . . . . 16 ⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → 𝑦𝑅𝑥)
39		rsp 3226	. . . . . . . . . . . . . . . 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 3446	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ∈ V
42		bnj1417.2	. . . . . . . . . . . . . . . . 17 ⊢ (𝜓 ↔ ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))
43		eleq1w 2820	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ trCl(𝑥, 𝐴, 𝑅) ↔ 𝑦 ∈ trCl(𝑥, 𝐴, 𝑅)))
44		bnj1318 35207	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑦 → trCl(𝑥, 𝐴, 𝑅) = trCl(𝑦, 𝐴, 𝑅))
45	44	eleq2d 2823	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → (𝑦 ∈ trCl(𝑥, 𝐴, 𝑅) ↔ 𝑦 ∈ trCl(𝑦, 𝐴, 𝑅)))
46	43, 45	bitrd 279	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ trCl(𝑥, 𝐴, 𝑅) ↔ 𝑦 ∈ trCl(𝑦, 𝐴, 𝑅)))
47	46	notbid 318	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅) ↔ ¬ 𝑦 ∈ trCl(𝑦, 𝐴, 𝑅)))
48	42, 47	bitrid 283	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ ¬ 𝑦 ∈ trCl(𝑦, 𝐴, 𝑅)))
49	41, 48	sbcie 3784	. . . . . . . . . . . . . . 15 ⊢ ([𝑦 / 𝑥]𝜓 ↔ ¬ 𝑦 ∈ trCl(𝑦, 𝐴, 𝑅))
50	40, 49	sylib 218	. . . . . . . . . . . . . 14 ⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → ¬ 𝑦 ∈ trCl(𝑦, 𝐴, 𝑅))
51	33, 50	pm2.65da 817	. . . . . . . . . . . . 13 ⊢ ((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → ¬ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅))
52	51	ex 412	. . . . . . . . . . . 12 ⊢ (𝜃 → (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → ¬ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)))
53	21, 52	ralrimi 3236	. . . . . . . . . . 11 ⊢ (𝜃 → ∀𝑦 ∈ pred (𝑥, 𝐴, 𝑅) ¬ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅))
54		ralnex 3064	. . . . . . . . . . 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 218	. . . . . . . . . 10 ⊢ (𝜃 → ¬ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝑥 ∈ trCl(𝑦, 𝐴, 𝑅))
56		eliun 4952	. . . . . . . . . 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 329	. . . . . . . . 9 ⊢ (𝜃 → ¬ 𝑥 ∈ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))
58		ioran 986	. . . . . . . . 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 584	. . . . . . . 8 ⊢ (𝜃 → ¬ (𝑥 ∈ pred(𝑥, 𝐴, 𝑅) ∨ 𝑥 ∈ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)))
60	3	simp2bi 1147	. . . . . . . . . . 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 35219	. . . . . . . . . . 11 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → trCl(𝑥, 𝐴, 𝑅) = 𝐵)
63	6, 60, 62	syl2anc 585	. . . . . . . . . 10 ⊢ (𝜃 → trCl(𝑥, 𝐴, 𝑅) = 𝐵)
64	63	eleq2d 2823	. . . . . . . . 9 ⊢ (𝜃 → (𝑥 ∈ trCl(𝑥, 𝐴, 𝑅) ↔ 𝑥 ∈ 𝐵))
65	61	bnj1138 34971	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵 ↔ (𝑥 ∈ pred(𝑥, 𝐴, 𝑅) ∨ 𝑥 ∈ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)))
66	64, 65	bitrdi 287	. . . . . . . 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 325	. . . . . . 7 ⊢ (𝜃 → ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))
68	67, 42	sylibr 234	. . . . . 6 ⊢ (𝜃 → 𝜓)
69	3, 68	sylbir 235	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝜒) → 𝜓)
70	69	3exp 1120	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜒 → 𝜓)))
71	70	ralrimiv 3129	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜒 → 𝜓))
72	17	bnj1204 35194	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜒 → 𝜓)) → ∀𝑥 ∈ 𝐴 𝜓)
73	2, 71, 72	syl2anc 585	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)
74	42	ralbii 3084	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))
75	73, 74	sylib 218	1 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∀wral 3052 ∃wrex 3062 [wsbc 3742 ∪ cun 3901 ⊆ wss 3903 ∪ ciun 4948 class class class wbr 5100 Fr wfr 5584 predc-bnj14 34871 Se w-bnj13 34873 FrSe w-bnj15 34875 trClc-bnj18 34877

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 2709 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5314 ax-pr 5381 ax-un 7692 ax-reg 9511 ax-inf2 9564

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 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5529 df-eprel 5534 df-po 5542 df-so 5543 df-fr 5587 df-we 5589 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-ord 6330 df-on 6331 df-lim 6332 df-suc 6333 df-iota 6458 df-fun 6504 df-fn 6505 df-f 6506 df-f1 6507 df-fo 6508 df-f1o 6509 df-fv 6510 df-om 7821 df-1o 8409 df-bnj17 34870 df-bnj14 34872 df-bnj13 34874 df-bnj15 34876 df-bnj18 34878 df-bnj19 34880

This theorem is referenced by: bnj1421 35224

W3C validator