bnj1417 - Mathbox for Jonathan Ben-Naim

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Theorem bnj1417 31923

Description: Technical lemma for bnj60 31944. 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 217	. . 3 ⊢ (𝜑 → 𝑅 FrSe 𝐴)
3		bnj1417.4	. . . . . 6 ⊢ (𝜃 ↔ (𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝜒))
4		bnj1418 31922	. . . . . . . . . . 11 ⊢ (𝑥 ∈ pred(𝑥, 𝐴, 𝑅) → 𝑥𝑅𝑥)
5	4	adantl 482	. . . . . . . . . 10 ⊢ ((𝜃 ∧ 𝑥 ∈ pred(𝑥, 𝐴, 𝑅)) → 𝑥𝑅𝑥)
6	3, 2	bnj835 31643	. . . . . . . . . . . 12 ⊢ (𝜃 → 𝑅 FrSe 𝐴)
7		df-bnj15 31576	. . . . . . . . . . . . 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 31766	. . . . . . . . . . . 12 ⊢ pred(𝑥, 𝐴, 𝑅) ⊆ 𝐴
11	10	sseli 3891	. . . . . . . . . . 11 ⊢ (𝑥 ∈ pred(𝑥, 𝐴, 𝑅) → 𝑥 ∈ 𝐴)
12		frirr 5427	. . . . . . . . . . 11 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥𝑅𝑥)
13	9, 11, 12	syl2an 595	. . . . . . . . . 10 ⊢ ((𝜃 ∧ 𝑥 ∈ pred(𝑥, 𝐴, 𝑅)) → ¬ 𝑥𝑅𝑥)
14	5, 13	pm2.65da 813	. . . . . . . . 9 ⊢ (𝜃 → ¬ 𝑥 ∈ pred(𝑥, 𝐴, 𝑅))
15		nfv 1896	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦𝜑
16		nfv 1896	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴
17		bnj1417.3	. . . . . . . . . . . . . . . 16 ⊢ (𝜒 ↔ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → [𝑦 / 𝑥]𝜓))
18	17	bnj1095 31666	. . . . . . . . . . . . . . 15 ⊢ (𝜒 → ∀𝑦𝜒)
19	18	nf5i 2119	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦𝜒
20	15, 16, 19	nf3an 1887	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦(𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝜒)
21	3, 20	nfxfr 1838	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦𝜃
22	6	ad2antrr 722	. . . . . . . . . . . . . . . 16 ⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → 𝑅 FrSe 𝐴)
23		simplr 765	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → 𝑦 ∈ pred(𝑥, 𝐴, 𝑅))
24	10, 23	sseldi 3893	. . . . . . . . . . . . . . . 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 31874	. . . . . . . . . . . . . . . 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 1364	. . . . . . . . . . . . . . 15 ⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → trCl(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑦, 𝐴, 𝑅))
28		bnj1147 31876	. . . . . . . . . . . . . . . . . 18 ⊢ trCl(𝑦, 𝐴, 𝑅) ⊆ 𝐴
29	28, 25	sseldi 3893	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → 𝑥 ∈ 𝐴)
30		bnj906 31814	. . . . . . . . . . . . . . . . 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 3896	. . . . . . . . . . . . . . 15 ⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → 𝑦 ∈ trCl(𝑥, 𝐴, 𝑅))
33	27, 32	sseldd 3896	. . . . . . . . . . . . . 14 ⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → 𝑦 ∈ trCl(𝑦, 𝐴, 𝑅))
34	17	biimpi 217	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜒 → ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → [𝑦 / 𝑥]𝜓))
35	3, 34	bnj837 31645	. . . . . . . . . . . . . . . . 17 ⊢ (𝜃 → ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → [𝑦 / 𝑥]𝜓))
36	35	ad2antrr 722	. . . . . . . . . . . . . . . 16 ⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → [𝑦 / 𝑥]𝜓))
37		bnj1418 31922	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → 𝑦𝑅𝑥)
38	37	ad2antlr 723	. . . . . . . . . . . . . . . 16 ⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → 𝑦𝑅𝑥)
39		rsp 3174	. . . . . . . . . . . . . . . 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 3443	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ∈ V
42		bnj1417.2	. . . . . . . . . . . . . . . . 17 ⊢ (𝜓 ↔ ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))
43		eleq1w 2867	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ trCl(𝑥, 𝐴, 𝑅) ↔ 𝑦 ∈ trCl(𝑥, 𝐴, 𝑅)))
44		bnj1318 31907	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑦 → trCl(𝑥, 𝐴, 𝑅) = trCl(𝑦, 𝐴, 𝑅))
45	44	eleq2d 2870	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → (𝑦 ∈ trCl(𝑥, 𝐴, 𝑅) ↔ 𝑦 ∈ trCl(𝑦, 𝐴, 𝑅)))
46	43, 45	bitrd 280	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ trCl(𝑥, 𝐴, 𝑅) ↔ 𝑦 ∈ trCl(𝑦, 𝐴, 𝑅)))
47	46	notbid 319	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅) ↔ ¬ 𝑦 ∈ trCl(𝑦, 𝐴, 𝑅)))
48	42, 47	syl5bb 284	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ ¬ 𝑦 ∈ trCl(𝑦, 𝐴, 𝑅)))
49	41, 48	sbcie 3746	. . . . . . . . . . . . . . 15 ⊢ ([𝑦 / 𝑥]𝜓 ↔ ¬ 𝑦 ∈ trCl(𝑦, 𝐴, 𝑅))
50	40, 49	sylib 219	. . . . . . . . . . . . . 14 ⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → ¬ 𝑦 ∈ trCl(𝑦, 𝐴, 𝑅))
51	33, 50	pm2.65da 813	. . . . . . . . . . . . 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 3185	. . . . . . . . . . 11 ⊢ (𝜃 → ∀𝑦 ∈ pred (𝑥, 𝐴, 𝑅) ¬ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅))
54		ralnex 3202	. . . . . . . . . . 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 219	. . . . . . . . . 10 ⊢ (𝜃 → ¬ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝑥 ∈ trCl(𝑦, 𝐴, 𝑅))
56		eliun 4835	. . . . . . . . . 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 330	. . . . . . . . 9 ⊢ (𝜃 → ¬ 𝑥 ∈ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))
58		ioran 978	. . . . . . . . 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 1139	. . . . . . . . . . 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 31919	. . . . . . . . . . 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 2870	. . . . . . . . 9 ⊢ (𝜃 → (𝑥 ∈ trCl(𝑥, 𝐴, 𝑅) ↔ 𝑥 ∈ 𝐵))
65	61	bnj1138 31673	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵 ↔ (𝑥 ∈ pred(𝑥, 𝐴, 𝑅) ∨ 𝑥 ∈ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)))
66	64, 65	syl6bb 288	. . . . . . . 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 326	. . . . . . 7 ⊢ (𝜃 → ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))
68	67, 42	sylibr 235	. . . . . 6 ⊢ (𝜃 → 𝜓)
69	3, 68	sylbir 236	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝜒) → 𝜓)
70	69	3exp 1112	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜒 → 𝜓)))
71	70	ralrimiv 3150	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜒 → 𝜓))
72	17	bnj1204 31894	. . 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 3134	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))
75	73, 74	sylib 219	1 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 ∨ wo 842 ∧ w3a 1080 = wceq 1525 ∈ wcel 2083 ∀wral 3107 ∃wrex 3108 [wsbc 3711 ∪ cun 3863 ⊆ wss 3865 ∪ ciun 4831 class class class wbr 4968 Fr wfr 5406 predc-bnj14 31571 Se w-bnj13 31573 FrSe w-bnj15 31575 trClc-bnj18 31577

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-rep 5088 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 ax-reg 8909 ax-inf2 8957

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-fal 1538 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-ral 3112 df-rex 3113 df-reu 3114 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-iun 4833 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-om 7444 df-1o 7960 df-bnj17 31570 df-bnj14 31572 df-bnj13 31574 df-bnj15 31576 df-bnj18 31578 df-bnj19 31580

This theorem is referenced by: bnj1421 31924

W3C validator