Theorem dfwe2 7768

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 5608	. 2 ⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴))
2		df-so 5562	. . . 4 ⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
3		simpr 484	. . . . 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 5631	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 Fr 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ¬ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥))
7	6	3adantr3 1172	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 Fr 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ¬ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥))
8		breq2 5123	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑧 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝑧))
9	8	anbi2d 630	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑧 → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧)))
10	9	notbid 318	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → (¬ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ ¬ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧)))
11	7, 10	syl5ibcom 245	. . . . . . . . . . . . . . 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 7766	. . . . . . . . . . . . . . . . 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 228	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) ∧ 𝑧𝑅𝑥) → (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧 ∧ 𝑧𝑅𝑥))
17	16	ancoms 458	. . . . . . . . . . . . . . . . 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 415	. . . . . . . . . . . . . 14 ⊢ ((𝑅 Fr 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝑧𝑅𝑥 → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
21	5, 13, 20	3jaod 1431	. . . . . . . . . . . . 13 ⊢ ((𝑅 Fr 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥𝑅𝑧 ∨ 𝑥 = 𝑧 ∨ 𝑧𝑅𝑥) → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
22		frirr 5630	. . . . . . . . . . . . . 14 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥𝑅𝑥)
23	22	3ad2antr1 1189	. . . . . . . . . . . . 13 ⊢ ((𝑅 Fr 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ¬ 𝑥𝑅𝑥)
24	21, 23	jctild 525	. . . . . . . . . . . 12 ⊢ ((𝑅 Fr 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥𝑅𝑧 ∨ 𝑥 = 𝑧 ∨ 𝑧𝑅𝑥) → (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
25	24	ex 412	. . . . . . . . . . 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 1916	. . . . . . . . 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 1918	. . . . . . . 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 3182	. . . . . . . 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 3182	. . . . . . . 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 5123	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → (𝑥𝑅𝑦 ↔ 𝑥𝑅𝑧))
33		equequ2 2025	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑧))
34		breq1 5122	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → (𝑦𝑅𝑥 ↔ 𝑧𝑅𝑥))
35	32, 33, 34	3orbi123d 1437	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → ((𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ (𝑥𝑅𝑧 ∨ 𝑥 = 𝑧 ∨ 𝑧𝑅𝑥)))
36	35	ralidmw 4483	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))
37	35	cbvralvw 3220	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑧 ∨ 𝑥 = 𝑧 ∨ 𝑧𝑅𝑥))
38	37	ralbii 3082	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑧 ∨ 𝑥 = 𝑧 ∨ 𝑧𝑅𝑥))
39	36, 38	bitr3i 277	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑧 ∨ 𝑥 = 𝑧 ∨ 𝑧𝑅𝑥))
40	39	ralbii 3082	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑧 ∨ 𝑥 = 𝑧 ∨ 𝑧𝑅𝑥))
41		df-po 5561	. . . . . . 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 551	. . . . 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 226	. . . 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 283	. . 3 ⊢ (𝑅 Fr 𝐴 → (𝑅 Or 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
46	45	pm5.32i 574	. 2 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴) ↔ (𝑅 Fr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
47	1, 46	bitri 275	1 ⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ w3o 1085   ∧ w3a 1086  ∀wal 1538   ∈ wcel 2108  ∀wral 3051   class class class wbr 5119   Po wpo 5559   Or wor 5560   Fr wfr 5603   We wwe 5605

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-tp 4606  df-op 4608  df-uni 4884  df-br 5120  df-po 5561  df-so 5562  df-fr 5606  df-we 5608

This theorem is referenced by:  epweonALT  7770  f1oweALT  7971  dford2  9634  fpwwe2lem11  10655  fpwwe2lem12  10656  dfon2  35810  fnwe2  43077