Theorem bnj1417 35238

Description: Technical lemma for bnj60 35259. 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 218	. . 3 ⊢ (𝜑 → 𝑅 FrSe 𝐴)
3		bnj1417.4	. . . . . 6 ⊢ (𝜃 ↔ (𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝜒))
4		bnj1418 35237	. . . . . . . . . . 11 ⊢ (𝑥 ∈ pred(𝑥, 𝐴, 𝑅) → 𝑥𝑅𝑥)
5	4	adantl 483	. . . . . . . . . 10 ⊢ ((𝜃 ∧ 𝑥 ∈ pred(𝑥, 𝐴, 𝑅)) → 𝑥𝑅𝑥)
6	3, 2	bnj835 34957	. . . . . . . . . . . 12 ⊢ (𝜃 → 𝑅 FrSe 𝐴)
7		df-bnj15 34891	. . . . . . . . . . . . 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 35079	. . . . . . . . . . . 12 ⊢ pred(𝑥, 𝐴, 𝑅) ⊆ 𝐴
11	10	sseli 3913	. . . . . . . . . . 11 ⊢ (𝑥 ∈ pred(𝑥, 𝐴, 𝑅) → 𝑥 ∈ 𝐴)
12		frirr 5597	. . . . . . . . . . 11 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥𝑅𝑥)
13	9, 11, 12	syl2an 603	. . . . . . . . . 10 ⊢ ((𝜃 ∧ 𝑥 ∈ pred(𝑥, 𝐴, 𝑅)) → ¬ 𝑥𝑅𝑥)
14	5, 13	pm2.65da 823	. . . . . . . . 9 ⊢ (𝜃 → ¬ 𝑥 ∈ pred(𝑥, 𝐴, 𝑅))
15		nfv 1922	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦𝜑
16		nfv 1922	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐴
17		bnj1417.3	. . . . . . . . . . . . . . . 16 ⊢ (𝜒 ↔ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → [𝑦 / 𝑥]𝜓))
18	17	bnj1095 34979	. . . . . . . . . . . . . . 15 ⊢ (𝜒 → ∀𝑦𝜒)
19	18	nf5i 2159	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑦𝜒
20	15, 16, 19	nf3an 1909	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑦(𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝜒)
21	3, 20	nfxfr 1861	. . . . . . . . . . . 12 ⊢ Ⅎ𝑦𝜃
22	6	ad2antrr 733	. . . . . . . . . . . . . . . 16 ⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → 𝑅 FrSe 𝐴)
23		simplr 775	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → 𝑦 ∈ pred(𝑥, 𝐴, 𝑅))
24	10, 23	sselid 3915	. . . . . . . . . . . . . . . 16 ⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → 𝑦 ∈ 𝐴)
25		simpr 486	. . . . . . . . . . . . . . . 16 ⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅))
26		bnj1125 35189	. . . . . . . . . . . . . . . 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 1380	. . . . . . . . . . . . . . 15 ⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → trCl(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑦, 𝐴, 𝑅))
28		bnj1147 35191	. . . . . . . . . . . . . . . . . 18 ⊢ trCl(𝑦, 𝐴, 𝑅) ⊆ 𝐴
29	28, 25	sselid 3915	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → 𝑥 ∈ 𝐴)
30		bnj906 35127	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
31	22, 29, 30	syl2anc 591	. . . . . . . . . . . . . . . 16 ⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
32	31, 23	sseldd 3918	. . . . . . . . . . . . . . 15 ⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → 𝑦 ∈ trCl(𝑥, 𝐴, 𝑅))
33	27, 32	sseldd 3918	. . . . . . . . . . . . . 14 ⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → 𝑦 ∈ trCl(𝑦, 𝐴, 𝑅))
34	17	biimpi 218	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜒 → ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → [𝑦 / 𝑥]𝜓))
35	3, 34	bnj837 34959	. . . . . . . . . . . . . . . . 17 ⊢ (𝜃 → ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → [𝑦 / 𝑥]𝜓))
36	35	ad2antrr 733	. . . . . . . . . . . . . . . 16 ⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → [𝑦 / 𝑥]𝜓))
37		bnj1418 35237	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → 𝑦𝑅𝑥)
38	37	ad2antlr 734	. . . . . . . . . . . . . . . 16 ⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → 𝑦𝑅𝑥)
39		rsp 3229	. . . . . . . . . . . . . . . 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 3437	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ∈ V
42		bnj1417.2	. . . . . . . . . . . . . . . . 17 ⊢ (𝜓 ↔ ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))
43		eleq1w 2824	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ trCl(𝑥, 𝐴, 𝑅) ↔ 𝑦 ∈ trCl(𝑥, 𝐴, 𝑅)))
44		bnj1318 35222	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑥 = 𝑦 → trCl(𝑥, 𝐴, 𝑅) = trCl(𝑦, 𝐴, 𝑅))
45	44	eleq2d 2827	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → (𝑦 ∈ trCl(𝑥, 𝐴, 𝑅) ↔ 𝑦 ∈ trCl(𝑦, 𝐴, 𝑅)))
46	43, 45	bitrd 281	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ trCl(𝑥, 𝐴, 𝑅) ↔ 𝑦 ∈ trCl(𝑦, 𝐴, 𝑅)))
47	46	notbid 320	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → (¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅) ↔ ¬ 𝑦 ∈ trCl(𝑦, 𝐴, 𝑅)))
48	42, 47	bitrid 285	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ ¬ 𝑦 ∈ trCl(𝑦, 𝐴, 𝑅)))
49	41, 48	sbcie 3766	. . . . . . . . . . . . . . 15 ⊢ ([𝑦 / 𝑥]𝜓 ↔ ¬ 𝑦 ∈ trCl(𝑦, 𝐴, 𝑅))
50	40, 49	sylib 220	. . . . . . . . . . . . . 14 ⊢ (((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) ∧ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)) → ¬ 𝑦 ∈ trCl(𝑦, 𝐴, 𝑅))
51	33, 50	pm2.65da 823	. . . . . . . . . . . . 13 ⊢ ((𝜃 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → ¬ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅))
52	51	ex 414	. . . . . . . . . . . 12 ⊢ (𝜃 → (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → ¬ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅)))
53	21, 52	ralrimi 3239	. . . . . . . . . . 11 ⊢ (𝜃 → ∀𝑦 ∈ pred (𝑥, 𝐴, 𝑅) ¬ 𝑥 ∈ trCl(𝑦, 𝐴, 𝑅))
54		ralnex 3067	. . . . . . . . . . 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 220	. . . . . . . . . 10 ⊢ (𝜃 → ¬ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝑥 ∈ trCl(𝑦, 𝐴, 𝑅))
56		eliun 4928	. . . . . . . . . 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 331	. . . . . . . . 9 ⊢ (𝜃 → ¬ 𝑥 ∈ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))
58		ioran 992	. . . . . . . . 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 590	. . . . . . . 8 ⊢ (𝜃 → ¬ (𝑥 ∈ pred(𝑥, 𝐴, 𝑅) ∨ 𝑥 ∈ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)))
60	3	simp2bi 1153	. . . . . . . . . . 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 35234	. . . . . . . . . . 11 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → trCl(𝑥, 𝐴, 𝑅) = 𝐵)
63	6, 60, 62	syl2anc 591	. . . . . . . . . 10 ⊢ (𝜃 → trCl(𝑥, 𝐴, 𝑅) = 𝐵)
64	63	eleq2d 2827	. . . . . . . . 9 ⊢ (𝜃 → (𝑥 ∈ trCl(𝑥, 𝐴, 𝑅) ↔ 𝑥 ∈ 𝐵))
65	61	bnj1138 34986	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝐵 ↔ (𝑥 ∈ pred(𝑥, 𝐴, 𝑅) ∨ 𝑥 ∈ ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)))
66	64, 65	bitrdi 289	. . . . . . . 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 327	. . . . . . 7 ⊢ (𝜃 → ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))
68	67, 42	sylibr 236	. . . . . 6 ⊢ (𝜃 → 𝜓)
69	3, 68	sylbir 237	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝜒) → 𝜓)
70	69	3exp 1126	. . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (𝜒 → 𝜓)))
71	70	ralrimiv 3132	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜒 → 𝜓))
72	17	bnj1204 35209	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜒 → 𝜓)) → ∀𝑥 ∈ 𝐴 𝜓)
73	2, 71, 72	syl2anc 591	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)
74	42	ralbii 3087	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))
75	73, 74	sylib 220	1 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 397   ∨ wo 854   ∧ w3a 1093   = wceq 1548   ∈ wcel 2121  ∀wral 3055  ∃wrex 3065  [wsbc 3725   ∪ cun 3883   ⊆ wss 3885  ∪ ciun 4924   class class class wbr 5075   Fr wfr 5571   predc-bnj14 34886   Se w-bnj13 34888   FrSe w-bnj15 34890   trClc-bnj18 34892

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682  ax-reg 9501  ax-inf2 9557

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-om 7811  df-1o 8399  df-bnj17 34885  df-bnj14 34887  df-bnj13 34889  df-bnj15 34891  df-bnj18 34893  df-bnj19 34895

This theorem is referenced by:  bnj1421  35239