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Theorem supmaxti 6476
 Description: The greatest element of a set is its supremum. Note that the converse is not true; the supremum might not be an element of the set considered. (Contributed by Jim Kingdon, 24-Nov-2021.)
Hypotheses
Ref Expression
supmaxti.ti ((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))
supmaxti.2 (𝜑𝐶𝐴)
supmaxti.3 (𝜑𝐶𝐵)
supmaxti.4 ((𝜑𝑦𝐵) → ¬ 𝐶𝑅𝑦)
Assertion
Ref Expression
supmaxti (𝜑 → sup(𝐵, 𝐴, 𝑅) = 𝐶)
Distinct variable groups:   𝑢,𝐴,𝑣,𝑦   𝑢,𝐵,𝑣,𝑦   𝑢,𝐶,𝑣,𝑦   𝑢,𝑅,𝑣,𝑦   𝜑,𝑢,𝑣,𝑦

Proof of Theorem supmaxti
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 supmaxti.ti . 2 ((𝜑 ∧ (𝑢𝐴𝑣𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))
2 supmaxti.2 . 2 (𝜑𝐶𝐴)
3 supmaxti.4 . 2 ((𝜑𝑦𝐵) → ¬ 𝐶𝑅𝑦)
4 supmaxti.3 . . 3 (𝜑𝐶𝐵)
5 simprr 499 . . 3 ((𝜑 ∧ (𝑦𝐴𝑦𝑅𝐶)) → 𝑦𝑅𝐶)
6 breq2 3797 . . . 4 (𝑥 = 𝐶 → (𝑦𝑅𝑥𝑦𝑅𝐶))
76rspcev 2702 . . 3 ((𝐶𝐵𝑦𝑅𝐶) → ∃𝑥𝐵 𝑦𝑅𝑥)
84, 5, 7syl2an2r 560 . 2 ((𝜑 ∧ (𝑦𝐴𝑦𝑅𝐶)) → ∃𝑥𝐵 𝑦𝑅𝑥)
91, 2, 3, 8eqsuptid 6469 1 (𝜑 → sup(𝐵, 𝐴, 𝑅) = 𝐶)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ↔ wb 103   = wceq 1285   ∈ wcel 1434  ∃wrex 2350   class class class wbr 3793  supcsup 6454 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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064 This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-reu 2356  df-rmo 2357  df-rab 2358  df-v 2604  df-sbc 2817  df-un 2978  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-iota 4897  df-riota 5499  df-sup 6456 This theorem is referenced by:  supsnti  6477  maxleim  10229
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