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Theorem wetriext 4578
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 4008 . . . . . 6 (𝑧 = 𝐵 → (𝑧𝑅𝐵𝐵𝑅𝐵))
2 breq1 4008 . . . . . 6 (𝑧 = 𝐵 → (𝑧𝑅𝐶𝐵𝑅𝐶))
31, 2bibi12d 235 . . . . 5 (𝑧 = 𝐵 → ((𝑧𝑅𝐵𝑧𝑅𝐶) ↔ (𝐵𝑅𝐵𝐵𝑅𝐶)))
4 wetriext.ext . . . . 5 (𝜑 → ∀𝑧𝐴 (𝑧𝑅𝐵𝑧𝑅𝐶))
5 wetriext.b . . . . 5 (𝜑𝐵𝐴)
63, 4, 5rspcdva 2848 . . . 4 (𝜑 → (𝐵𝑅𝐵𝐵𝑅𝐶))
76biimpar 297 . . 3 ((𝜑𝐵𝑅𝐶) → 𝐵𝑅𝐵)
8 wetriext.we . . . . . 6 (𝜑𝑅 We 𝐴)
9 wefr 4360 . . . . . 6 (𝑅 We 𝐴𝑅 Fr 𝐴)
108, 9syl 14 . . . . 5 (𝜑𝑅 Fr 𝐴)
11 wetriext.a . . . . 5 (𝜑𝐴𝑉)
12 frirrg 4352 . . . . 5 ((𝑅 Fr 𝐴𝐴𝑉𝐵𝐴) → ¬ 𝐵𝑅𝐵)
1310, 11, 5, 12syl3anc 1238 . . . 4 (𝜑 → ¬ 𝐵𝑅𝐵)
1413adantr 276 . . 3 ((𝜑𝐵𝑅𝐶) → ¬ 𝐵𝑅𝐵)
157, 14pm2.21dd 620 . 2 ((𝜑𝐵𝑅𝐶) → 𝐵 = 𝐶)
16 simpr 110 . 2 ((𝜑𝐵 = 𝐶) → 𝐵 = 𝐶)
17 breq1 4008 . . . . . 6 (𝑧 = 𝐶 → (𝑧𝑅𝐵𝐶𝑅𝐵))
18 breq1 4008 . . . . . 6 (𝑧 = 𝐶 → (𝑧𝑅𝐶𝐶𝑅𝐶))
1917, 18bibi12d 235 . . . . 5 (𝑧 = 𝐶 → ((𝑧𝑅𝐵𝑧𝑅𝐶) ↔ (𝐶𝑅𝐵𝐶𝑅𝐶)))
20 wetriext.c . . . . 5 (𝜑𝐶𝐴)
2119, 4, 20rspcdva 2848 . . . 4 (𝜑 → (𝐶𝑅𝐵𝐶𝑅𝐶))
2221biimpa 296 . . 3 ((𝜑𝐶𝑅𝐵) → 𝐶𝑅𝐶)
23 frirrg 4352 . . . . 5 ((𝑅 Fr 𝐴𝐴𝑉𝐶𝐴) → ¬ 𝐶𝑅𝐶)
2410, 11, 20, 23syl3anc 1238 . . . 4 (𝜑 → ¬ 𝐶𝑅𝐶)
2524adantr 276 . . 3 ((𝜑𝐶𝑅𝐵) → ¬ 𝐶𝑅𝐶)
2622, 25pm2.21dd 620 . 2 ((𝜑𝐶𝑅𝐵) → 𝐵 = 𝐶)
27 wetriext.tri . . 3 (𝜑 → ∀𝑎𝐴𝑏𝐴 (𝑎𝑅𝑏𝑎 = 𝑏𝑏𝑅𝑎))
28 breq1 4008 . . . . . 6 (𝑎 = 𝐵 → (𝑎𝑅𝑏𝐵𝑅𝑏))
29 eqeq1 2184 . . . . . 6 (𝑎 = 𝐵 → (𝑎 = 𝑏𝐵 = 𝑏))
30 breq2 4009 . . . . . 6 (𝑎 = 𝐵 → (𝑏𝑅𝑎𝑏𝑅𝐵))
3128, 29, 303orbi123d 1311 . . . . 5 (𝑎 = 𝐵 → ((𝑎𝑅𝑏𝑎 = 𝑏𝑏𝑅𝑎) ↔ (𝐵𝑅𝑏𝐵 = 𝑏𝑏𝑅𝐵)))
32 breq2 4009 . . . . . 6 (𝑏 = 𝐶 → (𝐵𝑅𝑏𝐵𝑅𝐶))
33 eqeq2 2187 . . . . . 6 (𝑏 = 𝐶 → (𝐵 = 𝑏𝐵 = 𝐶))
34 breq1 4008 . . . . . 6 (𝑏 = 𝐶 → (𝑏𝑅𝐵𝐶𝑅𝐵))
3532, 33, 343orbi123d 1311 . . . . 5 (𝑏 = 𝐶 → ((𝐵𝑅𝑏𝐵 = 𝑏𝑏𝑅𝐵) ↔ (𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵)))
3631, 35rspc2v 2856 . . . 4 ((𝐵𝐴𝐶𝐴) → (∀𝑎𝐴𝑏𝐴 (𝑎𝑅𝑏𝑎 = 𝑏𝑏𝑅𝑎) → (𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵)))
375, 20, 36syl2anc 411 . . 3 (𝜑 → (∀𝑎𝐴𝑏𝐴 (𝑎𝑅𝑏𝑎 = 𝑏𝑏𝑅𝑎) → (𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵)))
3827, 37mpd 13 . 2 (𝜑 → (𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵))
3915, 16, 26, 38mpjao3dan 1307 1 (𝜑𝐵 = 𝐶)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  w3o 977   = wceq 1353  wcel 2148  wral 2455   class class class wbr 4005   Fr wfr 4330   We wwe 4332
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-sep 4123
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-frfor 4333  df-frind 4334  df-wetr 4336
This theorem is referenced by: (None)
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