Theorem dfwe2 7624

Description: Alternate definition of well-ordering. Definition 6.24(2) of [TakeutiZaring] p. 30. (Contributed by NM, 16-Mar-1997.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)

Assertion

Ref	Expression
dfwe2	⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))

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

Proof of Theorem dfwe2

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-we 5546	. 2 ⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴))
2		df-so 5504	. . . 4 ⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
3		simpr 485	. . . . 5 ⊢ ((𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))
4		ax-1 6	. . . . . . . . . . . . . . 15 ⊢ (𝑥𝑅𝑧 → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
5	4	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝑅 Fr 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝑥𝑅𝑧 → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
6		fr2nr 5567	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 Fr 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ¬ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥))
7	6	3adantr3 1170	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 Fr 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ¬ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥))
8		breq2 5078	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑧 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝑧))
9	8	anbi2d 629	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑧 → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧)))
10	9	notbid 318	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → (¬ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ ¬ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧)))
11	7, 10	syl5ibcom 244	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 Fr 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝑥 = 𝑧 → ¬ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧)))
12		pm2.21 123	. . . . . . . . . . . . . . 15 ⊢ (¬ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
13	11, 12	syl6 35	. . . . . . . . . . . . . 14 ⊢ ((𝑅 Fr 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝑥 = 𝑧 → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
14		fr3nr 7622	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 Fr 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ¬ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧 ∧ 𝑧𝑅𝑥))
15		df-3an 1088	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧 ∧ 𝑧𝑅𝑥) ↔ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) ∧ 𝑧𝑅𝑥))
16	15	biimpri 227	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) ∧ 𝑧𝑅𝑥) → (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧 ∧ 𝑧𝑅𝑥))
17	16	ancoms 459	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧𝑅𝑥 ∧ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧)) → (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧 ∧ 𝑧𝑅𝑥))
18	14, 17	nsyl 140	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 Fr 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ¬ (𝑧𝑅𝑥 ∧ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧)))
19	18	pm2.21d 121	. . . . . . . . . . . . . . 15 ⊢ ((𝑅 Fr 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑧𝑅𝑥 ∧ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧)) → 𝑥𝑅𝑧))
20	19	expd 416	. . . . . . . . . . . . . 14 ⊢ ((𝑅 Fr 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝑧𝑅𝑥 → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
21	5, 13, 20	3jaod 1427	. . . . . . . . . . . . 13 ⊢ ((𝑅 Fr 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥𝑅𝑧 ∨ 𝑥 = 𝑧 ∨ 𝑧𝑅𝑥) → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
22		frirr 5566	. . . . . . . . . . . . . 14 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥𝑅𝑥)
23	22	3ad2antr1 1187	. . . . . . . . . . . . 13 ⊢ ((𝑅 Fr 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ¬ 𝑥𝑅𝑥)
24	21, 23	jctild 526	. . . . . . . . . . . 12 ⊢ ((𝑅 Fr 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥𝑅𝑧 ∨ 𝑥 = 𝑧 ∨ 𝑧𝑅𝑥) → (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
25	24	ex 413	. . . . . . . . . . 11 ⊢ (𝑅 Fr 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → ((𝑥𝑅𝑧 ∨ 𝑥 = 𝑧 ∨ 𝑧𝑅𝑥) → (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))))
26	25	a2d 29	. . . . . . . . . 10 ⊢ (𝑅 Fr 𝐴 → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑥𝑅𝑧 ∨ 𝑥 = 𝑧 ∨ 𝑧𝑅𝑥)) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))))
27	26	alimdv 1919	. . . . . . . . 9 ⊢ (𝑅 Fr 𝐴 → (∀𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑥𝑅𝑧 ∨ 𝑥 = 𝑧 ∨ 𝑧𝑅𝑥)) → ∀𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))))
28	27	2alimdv 1921	. . . . . . . 8 ⊢ (𝑅 Fr 𝐴 → (∀𝑥∀𝑦∀𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑥𝑅𝑧 ∨ 𝑥 = 𝑧 ∨ 𝑧𝑅𝑥)) → ∀𝑥∀𝑦∀𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))))
29		r3al 3119	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑧 ∨ 𝑥 = 𝑧 ∨ 𝑧𝑅𝑥) ↔ ∀𝑥∀𝑦∀𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑥𝑅𝑧 ∨ 𝑥 = 𝑧 ∨ 𝑧𝑅𝑥)))
30		r3al 3119	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ ∀𝑥∀𝑦∀𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
31	28, 29, 30	3imtr4g 296	. . . . . . 7 ⊢ (𝑅 Fr 𝐴 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑧 ∨ 𝑥 = 𝑧 ∨ 𝑧𝑅𝑥) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
32		breq2 5078	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → (𝑥𝑅𝑦 ↔ 𝑥𝑅𝑧))
33		equequ2 2029	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑧))
34		breq1 5077	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → (𝑦𝑅𝑥 ↔ 𝑧𝑅𝑥))
35	32, 33, 34	3orbi123d 1434	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → ((𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ (𝑥𝑅𝑧 ∨ 𝑥 = 𝑧 ∨ 𝑧𝑅𝑥)))
36	35	ralidmw 4438	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))
37	35	cbvralvw 3383	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑧 ∨ 𝑥 = 𝑧 ∨ 𝑧𝑅𝑥))
38	37	ralbii 3092	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑧 ∨ 𝑥 = 𝑧 ∨ 𝑧𝑅𝑥))
39	36, 38	bitr3i 276	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑧 ∨ 𝑥 = 𝑧 ∨ 𝑧𝑅𝑥))
40	39	ralbii 3092	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑧 ∨ 𝑥 = 𝑧 ∨ 𝑧𝑅𝑥))
41		df-po 5503	. . . . . . 7 ⊢ (𝑅 Po 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
42	31, 40, 41	3imtr4g 296	. . . . . 6 ⊢ (𝑅 Fr 𝐴 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) → 𝑅 Po 𝐴))
43	42	ancrd 552	. . . . 5 ⊢ (𝑅 Fr 𝐴 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) → (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))))
44	3, 43	impbid2 225	. . . 4 ⊢ (𝑅 Fr 𝐴 → ((𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
45	2, 44	bitrid 282	. . 3 ⊢ (𝑅 Fr 𝐴 → (𝑅 Or 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
46	45	pm5.32i 575	. 2 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴) ↔ (𝑅 Fr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
47	1, 46	bitri 274	1 ⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ w3o 1085   ∧ w3a 1086  ∀wal 1537   ∈ wcel 2106  ∀wral 3064   class class class wbr 5074   Po wpo 5501   Or wor 5502   Fr wfr 5541   We wwe 5543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4840  df-br 5075  df-po 5503  df-so 5504  df-fr 5544  df-we 5546

This theorem is referenced by:  epweonOLD  7626  f1oweALT  7815  dford2  9378  fpwwe2lem11  10397  fpwwe2lem12  10398  dfon2  33768  fnwe2  40878