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Theorem wetriext 4486
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 3927 . . . . . 6 (𝑧 = 𝐵 → (𝑧𝑅𝐵𝐵𝑅𝐵))
2 breq1 3927 . . . . . 6 (𝑧 = 𝐵 → (𝑧𝑅𝐶𝐵𝑅𝐶))
31, 2bibi12d 234 . . . . 5 (𝑧 = 𝐵 → ((𝑧𝑅𝐵𝑧𝑅𝐶) ↔ (𝐵𝑅𝐵𝐵𝑅𝐶)))
4 wetriext.ext . . . . 5 (𝜑 → ∀𝑧𝐴 (𝑧𝑅𝐵𝑧𝑅𝐶))
5 wetriext.b . . . . 5 (𝜑𝐵𝐴)
63, 4, 5rspcdva 2789 . . . 4 (𝜑 → (𝐵𝑅𝐵𝐵𝑅𝐶))
76biimpar 295 . . 3 ((𝜑𝐵𝑅𝐶) → 𝐵𝑅𝐵)
8 wetriext.we . . . . . 6 (𝜑𝑅 We 𝐴)
9 wefr 4275 . . . . . 6 (𝑅 We 𝐴𝑅 Fr 𝐴)
108, 9syl 14 . . . . 5 (𝜑𝑅 Fr 𝐴)
11 wetriext.a . . . . 5 (𝜑𝐴𝑉)
12 frirrg 4267 . . . . 5 ((𝑅 Fr 𝐴𝐴𝑉𝐵𝐴) → ¬ 𝐵𝑅𝐵)
1310, 11, 5, 12syl3anc 1216 . . . 4 (𝜑 → ¬ 𝐵𝑅𝐵)
1413adantr 274 . . 3 ((𝜑𝐵𝑅𝐶) → ¬ 𝐵𝑅𝐵)
157, 14pm2.21dd 609 . 2 ((𝜑𝐵𝑅𝐶) → 𝐵 = 𝐶)
16 simpr 109 . 2 ((𝜑𝐵 = 𝐶) → 𝐵 = 𝐶)
17 breq1 3927 . . . . . 6 (𝑧 = 𝐶 → (𝑧𝑅𝐵𝐶𝑅𝐵))
18 breq1 3927 . . . . . 6 (𝑧 = 𝐶 → (𝑧𝑅𝐶𝐶𝑅𝐶))
1917, 18bibi12d 234 . . . . 5 (𝑧 = 𝐶 → ((𝑧𝑅𝐵𝑧𝑅𝐶) ↔ (𝐶𝑅𝐵𝐶𝑅𝐶)))
20 wetriext.c . . . . 5 (𝜑𝐶𝐴)
2119, 4, 20rspcdva 2789 . . . 4 (𝜑 → (𝐶𝑅𝐵𝐶𝑅𝐶))
2221biimpa 294 . . 3 ((𝜑𝐶𝑅𝐵) → 𝐶𝑅𝐶)
23 frirrg 4267 . . . . 5 ((𝑅 Fr 𝐴𝐴𝑉𝐶𝐴) → ¬ 𝐶𝑅𝐶)
2410, 11, 20, 23syl3anc 1216 . . . 4 (𝜑 → ¬ 𝐶𝑅𝐶)
2524adantr 274 . . 3 ((𝜑𝐶𝑅𝐵) → ¬ 𝐶𝑅𝐶)
2622, 25pm2.21dd 609 . 2 ((𝜑𝐶𝑅𝐵) → 𝐵 = 𝐶)
27 wetriext.tri . . 3 (𝜑 → ∀𝑎𝐴𝑏𝐴 (𝑎𝑅𝑏𝑎 = 𝑏𝑏𝑅𝑎))
28 breq1 3927 . . . . . 6 (𝑎 = 𝐵 → (𝑎𝑅𝑏𝐵𝑅𝑏))
29 eqeq1 2144 . . . . . 6 (𝑎 = 𝐵 → (𝑎 = 𝑏𝐵 = 𝑏))
30 breq2 3928 . . . . . 6 (𝑎 = 𝐵 → (𝑏𝑅𝑎𝑏𝑅𝐵))
3128, 29, 303orbi123d 1289 . . . . 5 (𝑎 = 𝐵 → ((𝑎𝑅𝑏𝑎 = 𝑏𝑏𝑅𝑎) ↔ (𝐵𝑅𝑏𝐵 = 𝑏𝑏𝑅𝐵)))
32 breq2 3928 . . . . . 6 (𝑏 = 𝐶 → (𝐵𝑅𝑏𝐵𝑅𝐶))
33 eqeq2 2147 . . . . . 6 (𝑏 = 𝐶 → (𝐵 = 𝑏𝐵 = 𝐶))
34 breq1 3927 . . . . . 6 (𝑏 = 𝐶 → (𝑏𝑅𝐵𝐶𝑅𝐵))
3532, 33, 343orbi123d 1289 . . . . 5 (𝑏 = 𝐶 → ((𝐵𝑅𝑏𝐵 = 𝑏𝑏𝑅𝐵) ↔ (𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵)))
3631, 35rspc2v 2797 . . . 4 ((𝐵𝐴𝐶𝐴) → (∀𝑎𝐴𝑏𝐴 (𝑎𝑅𝑏𝑎 = 𝑏𝑏𝑅𝑎) → (𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵)))
375, 20, 36syl2anc 408 . . 3 (𝜑 → (∀𝑎𝐴𝑏𝐴 (𝑎𝑅𝑏𝑎 = 𝑏𝑏𝑅𝑎) → (𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵)))
3827, 37mpd 13 . 2 (𝜑 → (𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵))
3915, 16, 26, 38mpjao3dan 1285 1 (𝜑𝐵 = 𝐶)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  w3o 961   = wceq 1331  wcel 1480  wral 2414   class class class wbr 3924   Fr wfr 4245   We wwe 4247
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-frfor 4248  df-frind 4249  df-wetr 4251
This theorem is referenced by: (None)
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