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Theorem supmaxti 6753
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 500 . . 3 ((𝜑 ∧ (𝑦𝐴𝑦𝑅𝐶)) → 𝑦𝑅𝐶)
6 breq2 3855 . . . 4 (𝑥 = 𝐶 → (𝑦𝑅𝑥𝑦𝑅𝐶))
76rspcev 2723 . . 3 ((𝐶𝐵𝑦𝑅𝐶) → ∃𝑥𝐵 𝑦𝑅𝑥)
84, 5, 7syl2an2r 563 . 2 ((𝜑 ∧ (𝑦𝐴𝑦𝑅𝐶)) → ∃𝑥𝐵 𝑦𝑅𝑥)
91, 2, 3, 8eqsuptid 6746 1 (𝜑 → sup(𝐵, 𝐴, 𝑅) = 𝐶)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104   = wceq 1290  wcel 1439  wrex 2361   class class class wbr 3851  supcsup 6731
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-reu 2367  df-rmo 2368  df-rab 2369  df-v 2622  df-sbc 2842  df-un 3004  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-br 3852  df-iota 4993  df-riota 5622  df-sup 6733
This theorem is referenced by:  supsnti  6754  maxleim  10699  supfz  12188
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