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Theorem wetriext 4392
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 breq1 3848 . . . . . 6 (𝑧 = 𝐵 → (𝑧𝑅𝐵𝐵𝑅𝐵))
2 breq1 3848 . . . . . 6 (𝑧 = 𝐵 → (𝑧𝑅𝐶𝐵𝑅𝐶))
31, 2bibi12d 233 . . . . 5 (𝑧 = 𝐵 → ((𝑧𝑅𝐵𝑧𝑅𝐶) ↔ (𝐵𝑅𝐵𝐵𝑅𝐶)))
4 wetriext.ext . . . . 5 (𝜑 → ∀𝑧𝐴 (𝑧𝑅𝐵𝑧𝑅𝐶))
5 wetriext.b . . . . 5 (𝜑𝐵𝐴)
63, 4, 5rspcdva 2727 . . . 4 (𝜑 → (𝐵𝑅𝐵𝐵𝑅𝐶))
76biimpar 291 . . 3 ((𝜑𝐵𝑅𝐶) → 𝐵𝑅𝐵)
8 wetriext.we . . . . . 6 (𝜑𝑅 We 𝐴)
9 wefr 4185 . . . . . 6 (𝑅 We 𝐴𝑅 Fr 𝐴)
108, 9syl 14 . . . . 5 (𝜑𝑅 Fr 𝐴)
11 wetriext.a . . . . 5 (𝜑𝐴𝑉)
12 frirrg 4177 . . . . 5 ((𝑅 Fr 𝐴𝐴𝑉𝐵𝐴) → ¬ 𝐵𝑅𝐵)
1310, 11, 5, 12syl3anc 1174 . . . 4 (𝜑 → ¬ 𝐵𝑅𝐵)
1413adantr 270 . . 3 ((𝜑𝐵𝑅𝐶) → ¬ 𝐵𝑅𝐵)
157, 14pm2.21dd 585 . 2 ((𝜑𝐵𝑅𝐶) → 𝐵 = 𝐶)
16 simpr 108 . 2 ((𝜑𝐵 = 𝐶) → 𝐵 = 𝐶)
17 breq1 3848 . . . . . 6 (𝑧 = 𝐶 → (𝑧𝑅𝐵𝐶𝑅𝐵))
18 breq1 3848 . . . . . 6 (𝑧 = 𝐶 → (𝑧𝑅𝐶𝐶𝑅𝐶))
1917, 18bibi12d 233 . . . . 5 (𝑧 = 𝐶 → ((𝑧𝑅𝐵𝑧𝑅𝐶) ↔ (𝐶𝑅𝐵𝐶𝑅𝐶)))
20 wetriext.c . . . . 5 (𝜑𝐶𝐴)
2119, 4, 20rspcdva 2727 . . . 4 (𝜑 → (𝐶𝑅𝐵𝐶𝑅𝐶))
2221biimpa 290 . . 3 ((𝜑𝐶𝑅𝐵) → 𝐶𝑅𝐶)
23 frirrg 4177 . . . . 5 ((𝑅 Fr 𝐴𝐴𝑉𝐶𝐴) → ¬ 𝐶𝑅𝐶)
2410, 11, 20, 23syl3anc 1174 . . . 4 (𝜑 → ¬ 𝐶𝑅𝐶)
2524adantr 270 . . 3 ((𝜑𝐶𝑅𝐵) → ¬ 𝐶𝑅𝐶)
2622, 25pm2.21dd 585 . 2 ((𝜑𝐶𝑅𝐵) → 𝐵 = 𝐶)
27 wetriext.tri . . 3 (𝜑 → ∀𝑎𝐴𝑏𝐴 (𝑎𝑅𝑏𝑎 = 𝑏𝑏𝑅𝑎))
28 breq1 3848 . . . . . 6 (𝑎 = 𝐵 → (𝑎𝑅𝑏𝐵𝑅𝑏))
29 eqeq1 2094 . . . . . 6 (𝑎 = 𝐵 → (𝑎 = 𝑏𝐵 = 𝑏))
30 breq2 3849 . . . . . 6 (𝑎 = 𝐵 → (𝑏𝑅𝑎𝑏𝑅𝐵))
3128, 29, 303orbi123d 1247 . . . . 5 (𝑎 = 𝐵 → ((𝑎𝑅𝑏𝑎 = 𝑏𝑏𝑅𝑎) ↔ (𝐵𝑅𝑏𝐵 = 𝑏𝑏𝑅𝐵)))
32 breq2 3849 . . . . . 6 (𝑏 = 𝐶 → (𝐵𝑅𝑏𝐵𝑅𝐶))
33 eqeq2 2097 . . . . . 6 (𝑏 = 𝐶 → (𝐵 = 𝑏𝐵 = 𝐶))
34 breq1 3848 . . . . . 6 (𝑏 = 𝐶 → (𝑏𝑅𝐵𝐶𝑅𝐵))
3532, 33, 343orbi123d 1247 . . . . 5 (𝑏 = 𝐶 → ((𝐵𝑅𝑏𝐵 = 𝑏𝑏𝑅𝐵) ↔ (𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵)))
3631, 35rspc2v 2734 . . . 4 ((𝐵𝐴𝐶𝐴) → (∀𝑎𝐴𝑏𝐴 (𝑎𝑅𝑏𝑎 = 𝑏𝑏𝑅𝑎) → (𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵)))
375, 20, 36syl2anc 403 . . 3 (𝜑 → (∀𝑎𝐴𝑏𝐴 (𝑎𝑅𝑏𝑎 = 𝑏𝑏𝑅𝑎) → (𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵)))
3827, 37mpd 13 . 2 (𝜑 → (𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵))
3915, 16, 26, 38mpjao3dan 1243 1 (𝜑𝐵 = 𝐶)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  w3o 923   = wceq 1289  wcel 1438  wral 2359   class class class wbr 3845   Fr wfr 4155   We wwe 4157
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-v 2621  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-frfor 4158  df-frind 4159  df-wetr 4161
This theorem is referenced by: (None)
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