Theorem tratrb 42109

Description: If a class is transitive and any two distinct elements of the class are E-comparable, then every element of that class is transitive. Derived automatically from tratrbVD 42434. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
tratrb	⊢ ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) → Tr 𝐵)

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦

Proof of Theorem tratrb

Step	Hyp	Ref	Expression
1		nfv 1920	. . . 4 ⊢ Ⅎ𝑥Tr 𝐴
2		nfra1 3144	. . . 4 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦)
3		nfv 1920	. . . 4 ⊢ Ⅎ𝑥 𝐵 ∈ 𝐴
4	1, 2, 3	nf3an 1907	. . 3 ⊢ Ⅎ𝑥(Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴)
5		nfv 1920	. . . . 5 ⊢ Ⅎ𝑦Tr 𝐴
6		nfra2w 3153	. . . . 5 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦)
7		nfv 1920	. . . . 5 ⊢ Ⅎ𝑦 𝐵 ∈ 𝐴
8	5, 6, 7	nf3an 1907	. . . 4 ⊢ Ⅎ𝑦(Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴)
9		simpl 482	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝑦)
10	9	a1i 11	. . . . . . 7 ⊢ ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝑦))
11		simpr 484	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
12	11	a1i 11	. . . . . . 7 ⊢ ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵))
13		pm3.2an3 1338	. . . . . . 7 ⊢ (𝑥 ∈ 𝑦 → (𝑦 ∈ 𝐵 → (𝐵 ∈ 𝑥 → (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵 ∧ 𝐵 ∈ 𝑥))))
14	10, 12, 13	syl6c 70	. . . . . 6 ⊢ ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) → (𝐵 ∈ 𝑥 → (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵 ∧ 𝐵 ∈ 𝑥))))
15		en3lp 9333	. . . . . 6 ⊢ ¬ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵 ∧ 𝐵 ∈ 𝑥)
16		con3 153	. . . . . 6 ⊢ ((𝐵 ∈ 𝑥 → (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵 ∧ 𝐵 ∈ 𝑥)) → (¬ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵 ∧ 𝐵 ∈ 𝑥) → ¬ 𝐵 ∈ 𝑥))
17	14, 15, 16	syl6mpi 67	. . . . 5 ⊢ ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) → ¬ 𝐵 ∈ 𝑥))
18		eleq2 2828	. . . . . . . . 9 ⊢ (𝑥 = 𝐵 → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝐵))
19	18	biimprcd 249	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐵 → (𝑥 = 𝐵 → 𝑦 ∈ 𝑥))
20	12, 19	syl6 35	. . . . . . 7 ⊢ ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) → (𝑥 = 𝐵 → 𝑦 ∈ 𝑥)))
21		pm3.2 469	. . . . . . 7 ⊢ (𝑥 ∈ 𝑦 → (𝑦 ∈ 𝑥 → (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥)))
22	10, 20, 21	syl10 79	. . . . . 6 ⊢ ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) → (𝑥 = 𝐵 → (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥))))
23		en2lp 9325	. . . . . 6 ⊢ ¬ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥)
24		con3 153	. . . . . 6 ⊢ ((𝑥 = 𝐵 → (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥)) → (¬ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → ¬ 𝑥 = 𝐵))
25	22, 23, 24	syl6mpi 67	. . . . 5 ⊢ ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) → ¬ 𝑥 = 𝐵))
26		simp3 1136	. . . . . 6 ⊢ ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) → 𝐵 ∈ 𝐴)
27		simp1 1134	. . . . . . . . 9 ⊢ ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) → Tr 𝐴)
28		trel 5202	. . . . . . . . . . 11 ⊢ (Tr 𝐴 → ((𝑦 ∈ 𝐵 ∧ 𝐵 ∈ 𝐴) → 𝑦 ∈ 𝐴))
29	28	expd 415	. . . . . . . . . 10 ⊢ (Tr 𝐴 → (𝑦 ∈ 𝐵 → (𝐵 ∈ 𝐴 → 𝑦 ∈ 𝐴)))
30	27, 12, 26, 29	ee121 42078	. . . . . . . . 9 ⊢ ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐴))
31		trel 5202	. . . . . . . . . 10 ⊢ (Tr 𝐴 → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑥 ∈ 𝐴))
32	31	expd 415	. . . . . . . . 9 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝑦 → (𝑦 ∈ 𝐴 → 𝑥 ∈ 𝐴)))
33	27, 10, 30, 32	ee122 42079	. . . . . . . 8 ⊢ ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝐴))
34		ralcom 3282	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ↔ ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
35	34	biimpi 215	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) → ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
36	35	3ad2ant2 1132	. . . . . . . 8 ⊢ ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) → ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
37		rspsbc2 42107	. . . . . . . 8 ⊢ (𝐵 ∈ 𝐴 → (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) → [𝑥 / 𝑥][𝐵 / 𝑦](𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))))
38	26, 33, 36, 37	ee121 42078	. . . . . . 7 ⊢ ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) → [𝑥 / 𝑥][𝐵 / 𝑦](𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦)))
39		equid 2018	. . . . . . . 8 ⊢ 𝑥 = 𝑥
40		sbceq1a 3730	. . . . . . . 8 ⊢ (𝑥 = 𝑥 → ([𝐵 / 𝑦](𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ↔ [𝑥 / 𝑥][𝐵 / 𝑦](𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦)))
41	39, 40	ax-mp 5	. . . . . . 7 ⊢ ([𝐵 / 𝑦](𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ↔ [𝑥 / 𝑥][𝐵 / 𝑦](𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦))
42	38, 41	syl6ibr 251	. . . . . 6 ⊢ ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) → [𝐵 / 𝑦](𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦)))
43		sbcoreleleq 42108	. . . . . . 7 ⊢ (𝐵 ∈ 𝐴 → ([𝐵 / 𝑦](𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ↔ (𝑥 ∈ 𝐵 ∨ 𝐵 ∈ 𝑥 ∨ 𝑥 = 𝐵)))
44	43	biimpd 228	. . . . . 6 ⊢ (𝐵 ∈ 𝐴 → ([𝐵 / 𝑦](𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) → (𝑥 ∈ 𝐵 ∨ 𝐵 ∈ 𝑥 ∨ 𝑥 = 𝐵)))
45	26, 42, 44	sylsyld 61	. . . . 5 ⊢ ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) → (𝑥 ∈ 𝐵 ∨ 𝐵 ∈ 𝑥 ∨ 𝑥 = 𝐵)))
46		3ornot23 42082	. . . . . 6 ⊢ ((¬ 𝐵 ∈ 𝑥 ∧ ¬ 𝑥 = 𝐵) → ((𝑥 ∈ 𝐵 ∨ 𝐵 ∈ 𝑥 ∨ 𝑥 = 𝐵) → 𝑥 ∈ 𝐵))
47	46	ex 412	. . . . 5 ⊢ (¬ 𝐵 ∈ 𝑥 → (¬ 𝑥 = 𝐵 → ((𝑥 ∈ 𝐵 ∨ 𝐵 ∈ 𝑥 ∨ 𝑥 = 𝐵) → 𝑥 ∈ 𝐵)))
48	17, 25, 45, 47	ee222 42075	. . . 4 ⊢ ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝐵))
49	8, 48	alrimi 2209	. . 3 ⊢ ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) → ∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝐵))
50	4, 49	alrimi 2209	. 2 ⊢ ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) → ∀𝑥∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝐵))
51		dftr2 5197	. 2 ⊢ (Tr 𝐵 ↔ ∀𝑥∀𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝐵))
52	50, 51	sylibr 233	1 ⊢ ((Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑦 ∈ 𝑥 ∨ 𝑥 = 𝑦) ∧ 𝐵 ∈ 𝐴) → Tr 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ w3o 1084   ∧ w3a 1085  ∀wal 1539   = wceq 1541   ∈ wcel 2109  ∀wral 3065  [wsbc 3719  Tr wtr 5195

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pr 5355  ax-un 7579  ax-reg 9312

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-sbc 3720  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4845  df-br 5079  df-opab 5141  df-tr 5196  df-eprel 5494  df-fr 5543

This theorem is referenced by:  ordelordALT  42110  ordelordALTVD  42440