Theorem wetriext 4561

Description: A trichotomous well-order is extensional. (Contributed by Jim Kingdon, 26-Sep-2021.)

Hypotheses

Ref	Expression
wetriext.we	⊢ (𝜑 → 𝑅 We 𝐴)
wetriext.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
wetriext.tri	⊢ (𝜑 → ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑅𝑎))
wetriext.b	⊢ (𝜑 → 𝐵 ∈ 𝐴)
wetriext.c	⊢ (𝜑 → 𝐶 ∈ 𝐴)
wetriext.ext	⊢ (𝜑 → ∀𝑧 ∈ 𝐴 (𝑧𝑅𝐵 ↔ 𝑧𝑅𝐶))

Assertion

Ref	Expression
wetriext	⊢ (𝜑 → 𝐵 = 𝐶)

Distinct variable groups:   𝐴,𝑎,𝑏   𝑧,𝐴   𝐵,𝑎,𝑏   𝑧,𝐵   𝐶,𝑏   𝑧,𝐶   𝑅,𝑎,𝑏   𝑧,𝑅

Allowed substitution hints:   𝜑(𝑧,𝑎,𝑏)   𝐶(𝑎)   𝑉(𝑧,𝑎,𝑏)

Proof of Theorem wetriext

Step	Hyp	Ref	Expression
1		breq1 3992	. . . . . 6 ⊢ (𝑧 = 𝐵 → (𝑧𝑅𝐵 ↔ 𝐵𝑅𝐵))
2		breq1 3992	. . . . . 6 ⊢ (𝑧 = 𝐵 → (𝑧𝑅𝐶 ↔ 𝐵𝑅𝐶))
3	1, 2	bibi12d 234	. . . . 5 ⊢ (𝑧 = 𝐵 → ((𝑧𝑅𝐵 ↔ 𝑧𝑅𝐶) ↔ (𝐵𝑅𝐵 ↔ 𝐵𝑅𝐶)))
4		wetriext.ext	. . . . 5 ⊢ (𝜑 → ∀𝑧 ∈ 𝐴 (𝑧𝑅𝐵 ↔ 𝑧𝑅𝐶))
5		wetriext.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
6	3, 4, 5	rspcdva 2839	. . . 4 ⊢ (𝜑 → (𝐵𝑅𝐵 ↔ 𝐵𝑅𝐶))
7	6	biimpar 295	. . 3 ⊢ ((𝜑 ∧ 𝐵𝑅𝐶) → 𝐵𝑅𝐵)
8		wetriext.we	. . . . . 6 ⊢ (𝜑 → 𝑅 We 𝐴)
9		wefr 4343	. . . . . 6 ⊢ (𝑅 We 𝐴 → 𝑅 Fr 𝐴)
10	8, 9	syl 14	. . . . 5 ⊢ (𝜑 → 𝑅 Fr 𝐴)
11		wetriext.a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
12		frirrg 4335	. . . . 5 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵𝑅𝐵)
13	10, 11, 5, 12	syl3anc 1233	. . . 4 ⊢ (𝜑 → ¬ 𝐵𝑅𝐵)
14	13	adantr 274	. . 3 ⊢ ((𝜑 ∧ 𝐵𝑅𝐶) → ¬ 𝐵𝑅𝐵)
15	7, 14	pm2.21dd 615	. 2 ⊢ ((𝜑 ∧ 𝐵𝑅𝐶) → 𝐵 = 𝐶)
16		simpr 109	. 2 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝐵 = 𝐶)
17		breq1 3992	. . . . . 6 ⊢ (𝑧 = 𝐶 → (𝑧𝑅𝐵 ↔ 𝐶𝑅𝐵))
18		breq1 3992	. . . . . 6 ⊢ (𝑧 = 𝐶 → (𝑧𝑅𝐶 ↔ 𝐶𝑅𝐶))
19	17, 18	bibi12d 234	. . . . 5 ⊢ (𝑧 = 𝐶 → ((𝑧𝑅𝐵 ↔ 𝑧𝑅𝐶) ↔ (𝐶𝑅𝐵 ↔ 𝐶𝑅𝐶)))
20		wetriext.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
21	19, 4, 20	rspcdva 2839	. . . 4 ⊢ (𝜑 → (𝐶𝑅𝐵 ↔ 𝐶𝑅𝐶))
22	21	biimpa 294	. . 3 ⊢ ((𝜑 ∧ 𝐶𝑅𝐵) → 𝐶𝑅𝐶)
23		frirrg 4335	. . . . 5 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝐴) → ¬ 𝐶𝑅𝐶)
24	10, 11, 20, 23	syl3anc 1233	. . . 4 ⊢ (𝜑 → ¬ 𝐶𝑅𝐶)
25	24	adantr 274	. . 3 ⊢ ((𝜑 ∧ 𝐶𝑅𝐵) → ¬ 𝐶𝑅𝐶)
26	22, 25	pm2.21dd 615	. 2 ⊢ ((𝜑 ∧ 𝐶𝑅𝐵) → 𝐵 = 𝐶)
27		wetriext.tri	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑅𝑎))
28		breq1 3992	. . . . . 6 ⊢ (𝑎 = 𝐵 → (𝑎𝑅𝑏 ↔ 𝐵𝑅𝑏))
29		eqeq1 2177	. . . . . 6 ⊢ (𝑎 = 𝐵 → (𝑎 = 𝑏 ↔ 𝐵 = 𝑏))
30		breq2 3993	. . . . . 6 ⊢ (𝑎 = 𝐵 → (𝑏𝑅𝑎 ↔ 𝑏𝑅𝐵))
31	28, 29, 30	3orbi123d 1306	. . . . 5 ⊢ (𝑎 = 𝐵 → ((𝑎𝑅𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑅𝑎) ↔ (𝐵𝑅𝑏 ∨ 𝐵 = 𝑏 ∨ 𝑏𝑅𝐵)))
32		breq2 3993	. . . . . 6 ⊢ (𝑏 = 𝐶 → (𝐵𝑅𝑏 ↔ 𝐵𝑅𝐶))
33		eqeq2 2180	. . . . . 6 ⊢ (𝑏 = 𝐶 → (𝐵 = 𝑏 ↔ 𝐵 = 𝐶))
34		breq1 3992	. . . . . 6 ⊢ (𝑏 = 𝐶 → (𝑏𝑅𝐵 ↔ 𝐶𝑅𝐵))
35	32, 33, 34	3orbi123d 1306	. . . . 5 ⊢ (𝑏 = 𝐶 → ((𝐵𝑅𝑏 ∨ 𝐵 = 𝑏 ∨ 𝑏𝑅𝐵) ↔ (𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵)))
36	31, 35	rspc2v 2847	. . . 4 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑅𝑎) → (𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵)))
37	5, 20, 36	syl2anc 409	. . 3 ⊢ (𝜑 → (∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑅𝑎) → (𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵)))
38	27, 37	mpd 13	. 2 ⊢ (𝜑 → (𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵))
39	15, 16, 26, 38	mpjao3dan 1302	1 ⊢ (𝜑 → 𝐵 = 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ w3o 972   = wceq 1348   ∈ wcel 2141  ∀wral 2448   class class class wbr 3989   Fr wfr 4313   We wwe 4315

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-sep 4107

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-frfor 4316  df-frind 4317  df-wetr 4319

This theorem is referenced by: (None)