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Theorem wetriext 4329
Description: A trichotomous well-order is extensional. (Contributed by Jim Kingdon, 26-Sep-2021.)
Hypotheses
Ref Expression
wetriext.we (𝜑𝑅 We 𝐴)
wetriext.a (𝜑𝐴𝑉)
wetriext.tri (𝜑 → ∀𝑎𝐴𝑏𝐴 (𝑎𝑅𝑏𝑎 = 𝑏𝑏𝑅𝑎))
wetriext.b (𝜑𝐵𝐴)
wetriext.c (𝜑𝐶𝐴)
wetriext.ext (𝜑 → ∀𝑧𝐴 (𝑧𝑅𝐵𝑧𝑅𝐶))
Assertion
Ref Expression
wetriext (𝜑𝐵 = 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
StepHypRef Expression
1 wetriext.b . . . . 5 (𝜑𝐵𝐴)
2 wetriext.ext . . . . 5 (𝜑 → ∀𝑧𝐴 (𝑧𝑅𝐵𝑧𝑅𝐶))
3 breq1 3795 . . . . . . 7 (𝑧 = 𝐵 → (𝑧𝑅𝐵𝐵𝑅𝐵))
4 breq1 3795 . . . . . . 7 (𝑧 = 𝐵 → (𝑧𝑅𝐶𝐵𝑅𝐶))
53, 4bibi12d 228 . . . . . 6 (𝑧 = 𝐵 → ((𝑧𝑅𝐵𝑧𝑅𝐶) ↔ (𝐵𝑅𝐵𝐵𝑅𝐶)))
65rspcv 2669 . . . . 5 (𝐵𝐴 → (∀𝑧𝐴 (𝑧𝑅𝐵𝑧𝑅𝐶) → (𝐵𝑅𝐵𝐵𝑅𝐶)))
71, 2, 6sylc 60 . . . 4 (𝜑 → (𝐵𝑅𝐵𝐵𝑅𝐶))
87biimpar 285 . . 3 ((𝜑𝐵𝑅𝐶) → 𝐵𝑅𝐵)
9 wetriext.we . . . . . 6 (𝜑𝑅 We 𝐴)
10 wefr 4123 . . . . . 6 (𝑅 We 𝐴𝑅 Fr 𝐴)
119, 10syl 14 . . . . 5 (𝜑𝑅 Fr 𝐴)
12 wetriext.a . . . . 5 (𝜑𝐴𝑉)
13 frirrg 4115 . . . . 5 ((𝑅 Fr 𝐴𝐴𝑉𝐵𝐴) → ¬ 𝐵𝑅𝐵)
1411, 12, 1, 13syl3anc 1146 . . . 4 (𝜑 → ¬ 𝐵𝑅𝐵)
1514adantr 265 . . 3 ((𝜑𝐵𝑅𝐶) → ¬ 𝐵𝑅𝐵)
168, 15pm2.21dd 560 . 2 ((𝜑𝐵𝑅𝐶) → 𝐵 = 𝐶)
17 simpr 107 . 2 ((𝜑𝐵 = 𝐶) → 𝐵 = 𝐶)
18 wetriext.c . . . . 5 (𝜑𝐶𝐴)
19 breq1 3795 . . . . . . 7 (𝑧 = 𝐶 → (𝑧𝑅𝐵𝐶𝑅𝐵))
20 breq1 3795 . . . . . . 7 (𝑧 = 𝐶 → (𝑧𝑅𝐶𝐶𝑅𝐶))
2119, 20bibi12d 228 . . . . . 6 (𝑧 = 𝐶 → ((𝑧𝑅𝐵𝑧𝑅𝐶) ↔ (𝐶𝑅𝐵𝐶𝑅𝐶)))
2221rspcv 2669 . . . . 5 (𝐶𝐴 → (∀𝑧𝐴 (𝑧𝑅𝐵𝑧𝑅𝐶) → (𝐶𝑅𝐵𝐶𝑅𝐶)))
2318, 2, 22sylc 60 . . . 4 (𝜑 → (𝐶𝑅𝐵𝐶𝑅𝐶))
2423biimpa 284 . . 3 ((𝜑𝐶𝑅𝐵) → 𝐶𝑅𝐶)
25 frirrg 4115 . . . . 5 ((𝑅 Fr 𝐴𝐴𝑉𝐶𝐴) → ¬ 𝐶𝑅𝐶)
2611, 12, 18, 25syl3anc 1146 . . . 4 (𝜑 → ¬ 𝐶𝑅𝐶)
2726adantr 265 . . 3 ((𝜑𝐶𝑅𝐵) → ¬ 𝐶𝑅𝐶)
2824, 27pm2.21dd 560 . 2 ((𝜑𝐶𝑅𝐵) → 𝐵 = 𝐶)
29 wetriext.tri . . 3 (𝜑 → ∀𝑎𝐴𝑏𝐴 (𝑎𝑅𝑏𝑎 = 𝑏𝑏𝑅𝑎))
30 breq1 3795 . . . . . 6 (𝑎 = 𝐵 → (𝑎𝑅𝑏𝐵𝑅𝑏))
31 eqeq1 2062 . . . . . 6 (𝑎 = 𝐵 → (𝑎 = 𝑏𝐵 = 𝑏))
32 breq2 3796 . . . . . 6 (𝑎 = 𝐵 → (𝑏𝑅𝑎𝑏𝑅𝐵))
3330, 31, 323orbi123d 1217 . . . . 5 (𝑎 = 𝐵 → ((𝑎𝑅𝑏𝑎 = 𝑏𝑏𝑅𝑎) ↔ (𝐵𝑅𝑏𝐵 = 𝑏𝑏𝑅𝐵)))
34 breq2 3796 . . . . . 6 (𝑏 = 𝐶 → (𝐵𝑅𝑏𝐵𝑅𝐶))
35 eqeq2 2065 . . . . . 6 (𝑏 = 𝐶 → (𝐵 = 𝑏𝐵 = 𝐶))
36 breq1 3795 . . . . . 6 (𝑏 = 𝐶 → (𝑏𝑅𝐵𝐶𝑅𝐵))
3734, 35, 363orbi123d 1217 . . . . 5 (𝑏 = 𝐶 → ((𝐵𝑅𝑏𝐵 = 𝑏𝑏𝑅𝐵) ↔ (𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵)))
3833, 37rspc2v 2685 . . . 4 ((𝐵𝐴𝐶𝐴) → (∀𝑎𝐴𝑏𝐴 (𝑎𝑅𝑏𝑎 = 𝑏𝑏𝑅𝑎) → (𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵)))
391, 18, 38syl2anc 397 . . 3 (𝜑 → (∀𝑎𝐴𝑏𝐴 (𝑎𝑅𝑏𝑎 = 𝑏𝑏𝑅𝑎) → (𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵)))
4029, 39mpd 13 . 2 (𝜑 → (𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵))
4116, 17, 28, 40mpjao3dan 1213 1 (𝜑𝐵 = 𝐶)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wb 102  w3o 895   = wceq 1259  wcel 1409  wral 2323   class class class wbr 3792   Fr wfr 4093   We wwe 4095
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903
This theorem depends on definitions:  df-bi 114  df-3or 897  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-v 2576  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-frfor 4096  df-frind 4097  df-wetr 4099
This theorem is referenced by: (None)
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