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Theorem supmax 8256
 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 . . . 4 (𝜑𝐶𝐵)
54adantr 480 . . 3 ((𝜑 ∧ (𝑦𝐴𝑦𝑅𝐶)) → 𝐶𝐵)
6 simprr 792 . . 3 ((𝜑 ∧ (𝑦𝐴𝑦𝑅𝐶)) → 𝑦𝑅𝐶)
7 breq2 4587 . . . 4 (𝑧 = 𝐶 → (𝑦𝑅𝑧𝑦𝑅𝐶))
87rspcev 3282 . . 3 ((𝐶𝐵𝑦𝑅𝐶) → ∃𝑧𝐵 𝑦𝑅𝑧)
95, 6, 8syl2anc 691 . 2 ((𝜑 ∧ (𝑦𝐴𝑦𝑅𝐶)) → ∃𝑧𝐵 𝑦𝑅𝑧)
101, 2, 3, 9eqsupd 8246 1 (𝜑 → sup(𝐵, 𝐴, 𝑅) = 𝐶)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∃wrex 2897   class class class wbr 4583   Or wor 4958  supcsup 8229 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-po 4959  df-so 4960  df-iota 5768  df-riota 6511  df-sup 8231 This theorem is referenced by:  suppr  8260  lbinf  10855  gsumesum  29448  supfz  30866  inffzOLD  30868  mblfinlem2  32617
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