bnj1417 - Mathbox for Jonathan Ben-Naim

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Theorem bnj1417 34886

Description: Technical lemma for bnj60 34907. 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 34885	. . . . . . . . . . 11 ⊢ (𝑥 ∈ pred(𝑥, 𝐴, 𝑅) → 𝑥𝑅𝑥)
5	4	adantl 480	. . . . . . . . . 10 ⊢ ((𝜃 ∧ 𝑥 ∈ pred(𝑥, 𝐴, 𝑅)) → 𝑥𝑅𝑥)
6	3, 2	bnj835 34604	. . . . . . . . . . . 12 ⊢ (𝜃 → 𝑅 FrSe 𝐴)
7		df-bnj15 34538	. . . . . . . . . . . . 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 34727	. . . . . . . . . . . 12 ⊢ pred(𝑥, 𝐴, 𝑅) ⊆ 𝐴
11	10	sseli 3975	. . . . . . . . . . 11 ⊢ (𝑥 ∈ pred(𝑥, 𝐴, 𝑅) → 𝑥 ∈ 𝐴)
12		frirr 5659	. . . . . . . . . . 11 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥𝑅𝑥)
13	9, 11, 12	syl2an 594	. . . . . . . . . 10 ⊢ ((𝜃 ∧ 𝑥 ∈ pred(𝑥, 𝐴, 𝑅)) → ¬ 𝑥𝑅𝑥)
14	5, 13	pm2.65da 815	. . . . . . . . 9 ⊢ (𝜃 → ¬ 𝑥 ∈ pred(𝑥, 𝐴, 𝑅))
15		nfv 1910	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦𝜑
16		nfv 1910	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴
17		bnj1417.3	. . . . . . . . . . . . . . . 16 ⊢ (𝜒 ↔ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → [𝑦 / 𝑥]𝜓))
18	17	bnj1095 34626	. . . . . . . . . . . . . . 15 ⊢ (𝜒 → ∀𝑦𝜒)
19	18	nf5i 2135	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦𝜒
20	15, 16, 19	nf3an 1897	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦(𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝜒)
21	3, 20	nfxfr 1848	. . . . . . . . . . . 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 3977	. . . . . . . . . . . . . . . 16 ⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → 𝑦 ∈ 𝐴)
25		simpr 483	. . . . . . . . . . . . . . . 16 ⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅))
26		bnj1125 34837	. . . . . . . . . . . . . . . 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 1368	. . . . . . . . . . . . . . 15 ⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → trCl(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑦, 𝐴, 𝑅))
28		bnj1147 34839	. . . . . . . . . . . . . . . . . 18 ⊢ trCl(𝑦, 𝐴, 𝑅) ⊆ 𝐴
29	28, 25	sselid 3977	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → 𝑥 ∈ 𝐴)
30		bnj906 34775	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
31	22, 29, 30	syl2anc 582	. . . . . . . . . . . . . . . 16 ⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
32	31, 23	sseldd 3980	. . . . . . . . . . . . . . 15 ⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → 𝑦 ∈ trCl(𝑥, 𝐴, 𝑅))
33	27, 32	sseldd 3980	. . . . . . . . . . . . . 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 34606	. . . . . . . . . . . . . . . . 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 34885	. . . . . . . . . . . . . . . . 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 3235	. . . . . . . . . . . . . . . 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 3466	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ∈ V
42		bnj1417.2	. . . . . . . . . . . . . . . . 17 ⊢ (𝜓 ↔ ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))
43		eleq1w 2809	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ trCl(𝑥, 𝐴, 𝑅) ↔ 𝑦 ∈ trCl(𝑥, 𝐴, 𝑅)))
44		bnj1318 34870	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑦 → trCl(𝑥, 𝐴, 𝑅) = trCl(𝑦, 𝐴, 𝑅))
45	44	eleq2d 2812	. . . . . . . . . . . . . . . . . . 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 411	. . . . . . . . . . . 12 ⊢ (𝜃 → (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → ¬ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)))
53	21, 52	ralrimi 3245	. . . . . . . . . . 11 ⊢ (𝜃 → ∀𝑦 ∈ pred (𝑥, 𝐴, 𝑅) ¬ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅))
54		ralnex 3062	. . . . . . . . . . 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 5005	. . . . . . . . . 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 981	. . . . . . . . 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 581	. . . . . . . 8 ⊢ (𝜃 → ¬ (𝑥 ∈ pred(𝑥, 𝐴, 𝑅) ∨ 𝑥 ∈ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)))
60	3	simp2bi 1143	. . . . . . . . . . 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 34882	. . . . . . . . . . 11 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → trCl(𝑥, 𝐴, 𝑅) = 𝐵)
63	6, 60, 62	syl2anc 582	. . . . . . . . . 10 ⊢ (𝜃 → trCl(𝑥, 𝐴, 𝑅) = 𝐵)
64	63	eleq2d 2812	. . . . . . . . 9 ⊢ (𝜃 → (𝑥 ∈ trCl(𝑥, 𝐴, 𝑅) ↔ 𝑥 ∈ 𝐵))
65	61	bnj1138 34633	. . . . . . . . 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 1116	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜒 → 𝜓)))
71	70	ralrimiv 3135	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜒 → 𝜓))
72	17	bnj1204 34857	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜒 → 𝜓)) → ∀𝑥 ∈ 𝐴 𝜓)
73	2, 71, 72	syl2anc 582	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)
74	42	ralbii 3083	. 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 394 ∨ wo 845 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ∀wral 3051 ∃wrex 3060 [wsbc 3776 ∪ cun 3945 ⊆ wss 3947 ∪ ciun 5001 class class class wbr 5153 Fr wfr 5634 predc-bnj14 34533 Se w-bnj13 34535 FrSe w-bnj15 34537 trClc-bnj18 34539

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-rep 5290 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-reg 9635 ax-inf2 9684

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-tr 5271 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-ord 6379 df-on 6380 df-lim 6381 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-om 7877 df-1o 8496 df-bnj17 34532 df-bnj14 34534 df-bnj13 34536 df-bnj15 34538 df-bnj18 34540 df-bnj19 34542

This theorem is referenced by: bnj1421 34887

W3C validator