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Theorem supmax 8928
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 Jeff Hoffman, 17-Jun-2008.) (Proof shortened by OpenAI, 30-Mar-2020.)
Hypotheses
Ref Expression
supmax.1 (𝜑𝑅 Or 𝐴)
supmax.2 (𝜑𝐶𝐴)
supmax.3 (𝜑𝐶𝐵)
supmax.4 ((𝜑𝑦𝐵) → ¬ 𝐶𝑅𝑦)
Assertion
Ref Expression
supmax (𝜑 → sup(𝐵, 𝐴, 𝑅) = 𝐶)
Distinct variable groups:   𝑦,𝐴   𝑦,𝐵   𝑦,𝐶   𝑦,𝑅   𝜑,𝑦

Proof of Theorem supmax
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 supmax.1 . 2 (𝜑𝑅 Or 𝐴)
2 supmax.2 . 2 (𝜑𝐶𝐴)
3 supmax.4 . 2 ((𝜑𝑦𝐵) → ¬ 𝐶𝑅𝑦)
4 supmax.3 . . 3 (𝜑𝐶𝐵)
5 simprr 772 . . 3 ((𝜑 ∧ (𝑦𝐴𝑦𝑅𝐶)) → 𝑦𝑅𝐶)
6 breq2 5056 . . . 4 (𝑧 = 𝐶 → (𝑦𝑅𝑧𝑦𝑅𝐶))
76rspcev 3609 . . 3 ((𝐶𝐵𝑦𝑅𝐶) → ∃𝑧𝐵 𝑦𝑅𝑧)
84, 5, 7syl2an2r 684 . 2 ((𝜑 ∧ (𝑦𝐴𝑦𝑅𝐶)) → ∃𝑧𝐵 𝑦𝑅𝑧)
91, 2, 3, 8eqsupd 8918 1 (𝜑 → sup(𝐵, 𝐴, 𝑅) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1538  wcel 2115  wrex 3134   class class class wbr 5052   Or wor 5460  supcsup 8901
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-po 5461  df-so 5462  df-iota 6302  df-riota 7107  df-sup 8903
This theorem is referenced by:  suppr  8932  gsumesum  31378  supfz  33020  mblfinlem2  35043
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