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Theorem supmax 9464
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 769 . . 3 ((𝜑 ∧ (𝑦𝐴𝑦𝑅𝐶)) → 𝑦𝑅𝐶)
6 breq2 5151 . . . 4 (𝑧 = 𝐶 → (𝑦𝑅𝑧𝑦𝑅𝐶))
76rspcev 3611 . . 3 ((𝐶𝐵𝑦𝑅𝐶) → ∃𝑧𝐵 𝑦𝑅𝑧)
84, 5, 7syl2an2r 681 . 2 ((𝜑 ∧ (𝑦𝐴𝑦𝑅𝐶)) → ∃𝑧𝐵 𝑦𝑅𝑧)
91, 2, 3, 8eqsupd 9454 1 (𝜑 → sup(𝐵, 𝐴, 𝑅) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394   = wceq 1539  wcel 2104  wrex 3068   class class class wbr 5147   Or wor 5586  supcsup 9437
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-po 5587  df-so 5588  df-iota 6494  df-riota 7367  df-sup 9439
This theorem is referenced by:  suppr  9468  gsumesum  33355  supfz  35002  mblfinlem2  36829
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