Theorem dfwe2 7496

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 5516	. 2 ⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴))
2		df-so 5475	. . . 4 ⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
3		simpr 487	. . . . 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 5533	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 Fr 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ¬ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥))
7	6	3adantr3 1167	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 Fr 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ¬ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥))
8		breq2 5070	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑧 → (𝑦𝑅𝑥 ↔ 𝑦𝑅𝑧))
9	8	anbi2d 630	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑧 → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧)))
10	9	notbid 320	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → (¬ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ ¬ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧)))
11	7, 10	syl5ibcom 247	. . . . . . . . . . . . . . 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 7494	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅 Fr 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ¬ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧 ∧ 𝑧𝑅𝑥))
15		df-3an 1085	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧 ∧ 𝑧𝑅𝑥) ↔ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) ∧ 𝑧𝑅𝑥))
16	15	biimpri 230	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) ∧ 𝑧𝑅𝑥) → (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧 ∧ 𝑧𝑅𝑥))
17	16	ancoms 461	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑧𝑅𝑥 ∧ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧)) → (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧 ∧ 𝑧𝑅𝑥))
18	14, 17	nsyl 142	. . . . . . . . . . . . . . . 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 418	. . . . . . . . . . . . . 14 ⊢ ((𝑅 Fr 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝑧𝑅𝑥 → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
21	5, 13, 20	3jaod 1424	. . . . . . . . . . . . 13 ⊢ ((𝑅 Fr 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥𝑅𝑧 ∨ 𝑥 = 𝑧 ∨ 𝑧𝑅𝑥) → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
22		frirr 5532	. . . . . . . . . . . . . 14 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥𝑅𝑥)
23	22	3ad2antr1 1184	. . . . . . . . . . . . 13 ⊢ ((𝑅 Fr 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ¬ 𝑥𝑅𝑥)
24	21, 23	jctild 528	. . . . . . . . . . . 12 ⊢ ((𝑅 Fr 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥𝑅𝑧 ∨ 𝑥 = 𝑧 ∨ 𝑧𝑅𝑥) → (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
25	24	ex 415	. . . . . . . . . . 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 1917	. . . . . . . . 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 1919	. . . . . . . 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 3202	. . . . . . . 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 3202	. . . . . . . 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 298	. . . . . . 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		ralidm 4455	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))
33		breq2 5070	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → (𝑥𝑅𝑦 ↔ 𝑥𝑅𝑧))
34		equequ2 2033	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑧))
35		breq1 5069	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑧 → (𝑦𝑅𝑥 ↔ 𝑧𝑅𝑥))
36	33, 34, 35	3orbi123d 1431	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑧 → ((𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ (𝑥𝑅𝑧 ∨ 𝑥 = 𝑧 ∨ 𝑧𝑅𝑥)))
37	36	cbvralvw 3449	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑧 ∨ 𝑥 = 𝑧 ∨ 𝑧𝑅𝑥))
38	37	ralbii 3165	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑧 ∨ 𝑥 = 𝑧 ∨ 𝑧𝑅𝑥))
39	32, 38	bitr3i 279	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑧 ∨ 𝑥 = 𝑧 ∨ 𝑧𝑅𝑥))
40	39	ralbii 3165	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑧 ∨ 𝑥 = 𝑧 ∨ 𝑧𝑅𝑥))
41		df-po 5474	. . . . . . 7 ⊢ (𝑅 Po 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
42	31, 40, 41	3imtr4g 298	. . . . . 6 ⊢ (𝑅 Fr 𝐴 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) → 𝑅 Po 𝐴))
43	42	ancrd 554	. . . . 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 228	. . . 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	syl5bb 285	. . 3 ⊢ (𝑅 Fr 𝐴 → (𝑅 Or 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
46	45	pm5.32i 577	. 2 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴) ↔ (𝑅 Fr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
47	1, 46	bitri 277	1 ⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∨ w3o 1082   ∧ w3a 1083  ∀wal 1535   ∈ wcel 2114  ∀wral 3138   class class class wbr 5066   Po wpo 5472   Or wor 5473   Fr wfr 5511   We wwe 5513

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4839  df-br 5067  df-po 5474  df-so 5475  df-fr 5514  df-we 5516

This theorem is referenced by:  epweon  7497  f1oweALT  7673  dford2  9083  fpwwe2lem12  10063  fpwwe2lem13  10064  dfon2  33037  fnwe2  39673