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Theorem supmax 8540
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 813 . . 3 ((𝜑 ∧ (𝑦𝐴𝑦𝑅𝐶)) → 𝑦𝑅𝐶)
6 breq2 4808 . . . 4 (𝑧 = 𝐶 → (𝑦𝑅𝑧𝑦𝑅𝐶))
76rspcev 3449 . . 3 ((𝐶𝐵𝑦𝑅𝐶) → ∃𝑧𝐵 𝑦𝑅𝑧)
84, 5, 7syl2an2r 911 . 2 ((𝜑 ∧ (𝑦𝐴𝑦𝑅𝐶)) → ∃𝑧𝐵 𝑦𝑅𝑧)
91, 2, 3, 8eqsupd 8530 1 (𝜑 → sup(𝐵, 𝐴, 𝑅) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1632  wcel 2139  wrex 3051   class class class wbr 4804   Or wor 5186  supcsup 8513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-po 5187  df-so 5188  df-iota 6012  df-riota 6775  df-sup 8515
This theorem is referenced by:  suppr  8544  gsumesum  30451  supfz  31941  inffzOLD  31943  mblfinlem2  33778
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