Theorem wetriext 4588

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 4018	. . . . . 6 ⊢ (𝑧 = 𝐵 → (𝑧𝑅𝐵 ↔ 𝐵𝑅𝐵))
2		breq1 4018	. . . . . 6 ⊢ (𝑧 = 𝐵 → (𝑧𝑅𝐶 ↔ 𝐵𝑅𝐶))
3	1, 2	bibi12d 235	. . . . 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 2858	. . . 4 ⊢ (𝜑 → (𝐵𝑅𝐵 ↔ 𝐵𝑅𝐶))
7	6	biimpar 297	. . 3 ⊢ ((𝜑 ∧ 𝐵𝑅𝐶) → 𝐵𝑅𝐵)
8		wetriext.we	. . . . . 6 ⊢ (𝜑 → 𝑅 We 𝐴)
9		wefr 4370	. . . . . 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 4362	. . . . 5 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵𝑅𝐵)
13	10, 11, 5, 12	syl3anc 1248	. . . 4 ⊢ (𝜑 → ¬ 𝐵𝑅𝐵)
14	13	adantr 276	. . 3 ⊢ ((𝜑 ∧ 𝐵𝑅𝐶) → ¬ 𝐵𝑅𝐵)
15	7, 14	pm2.21dd 621	. 2 ⊢ ((𝜑 ∧ 𝐵𝑅𝐶) → 𝐵 = 𝐶)
16		simpr 110	. 2 ⊢ ((𝜑 ∧ 𝐵 = 𝐶) → 𝐵 = 𝐶)
17		breq1 4018	. . . . . 6 ⊢ (𝑧 = 𝐶 → (𝑧𝑅𝐵 ↔ 𝐶𝑅𝐵))
18		breq1 4018	. . . . . 6 ⊢ (𝑧 = 𝐶 → (𝑧𝑅𝐶 ↔ 𝐶𝑅𝐶))
19	17, 18	bibi12d 235	. . . . 5 ⊢ (𝑧 = 𝐶 → ((𝑧𝑅𝐵 ↔ 𝑧𝑅𝐶) ↔ (𝐶𝑅𝐵 ↔ 𝐶𝑅𝐶)))
20		wetriext.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
21	19, 4, 20	rspcdva 2858	. . . 4 ⊢ (𝜑 → (𝐶𝑅𝐵 ↔ 𝐶𝑅𝐶))
22	21	biimpa 296	. . 3 ⊢ ((𝜑 ∧ 𝐶𝑅𝐵) → 𝐶𝑅𝐶)
23		frirrg 4362	. . . . 5 ⊢ ((𝑅 Fr 𝐴 ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝐴) → ¬ 𝐶𝑅𝐶)
24	10, 11, 20, 23	syl3anc 1248	. . . 4 ⊢ (𝜑 → ¬ 𝐶𝑅𝐶)
25	24	adantr 276	. . 3 ⊢ ((𝜑 ∧ 𝐶𝑅𝐵) → ¬ 𝐶𝑅𝐶)
26	22, 25	pm2.21dd 621	. 2 ⊢ ((𝜑 ∧ 𝐶𝑅𝐵) → 𝐵 = 𝐶)
27		wetriext.tri	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑅𝑎))
28		breq1 4018	. . . . . 6 ⊢ (𝑎 = 𝐵 → (𝑎𝑅𝑏 ↔ 𝐵𝑅𝑏))
29		eqeq1 2194	. . . . . 6 ⊢ (𝑎 = 𝐵 → (𝑎 = 𝑏 ↔ 𝐵 = 𝑏))
30		breq2 4019	. . . . . 6 ⊢ (𝑎 = 𝐵 → (𝑏𝑅𝑎 ↔ 𝑏𝑅𝐵))
31	28, 29, 30	3orbi123d 1321	. . . . 5 ⊢ (𝑎 = 𝐵 → ((𝑎𝑅𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑅𝑎) ↔ (𝐵𝑅𝑏 ∨ 𝐵 = 𝑏 ∨ 𝑏𝑅𝐵)))
32		breq2 4019	. . . . . 6 ⊢ (𝑏 = 𝐶 → (𝐵𝑅𝑏 ↔ 𝐵𝑅𝐶))
33		eqeq2 2197	. . . . . 6 ⊢ (𝑏 = 𝐶 → (𝐵 = 𝑏 ↔ 𝐵 = 𝐶))
34		breq1 4018	. . . . . 6 ⊢ (𝑏 = 𝐶 → (𝑏𝑅𝐵 ↔ 𝐶𝑅𝐵))
35	32, 33, 34	3orbi123d 1321	. . . . 5 ⊢ (𝑏 = 𝐶 → ((𝐵𝑅𝑏 ∨ 𝐵 = 𝑏 ∨ 𝑏𝑅𝐵) ↔ (𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵)))
36	31, 35	rspc2v 2866	. . . 4 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (∀𝑎 ∈ 𝐴 ∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 ∨ 𝑎 = 𝑏 ∨ 𝑏𝑅𝑎) → (𝐵𝑅𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶𝑅𝐵)))
37	5, 20, 36	syl2anc 411	. . 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 1317	1 ⊢ (𝜑 → 𝐵 = 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ w3o 978   = wceq 1363   ∈ wcel 2158  ∀wral 2465   class class class wbr 4015   Fr wfr 4340   We wwe 4342

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169  ax-sep 4133

This theorem depends on definitions:  df-bi 117  df-3or 980  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-v 2751  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-frfor 4343  df-frind 4344  df-wetr 4346

This theorem is referenced by: (None)