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Theorem sup0 8996
Description: The supremum of an empty set under a base set which has a unique smallest element is the smallest element of the base set. (Contributed by AV, 4-Sep-2020.)
Assertion
Ref Expression
sup0 ((𝑅 Or 𝐴 ∧ (𝑋𝐴 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑋) ∧ ∃!𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥) → sup(∅, 𝐴, 𝑅) = 𝑋)
Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝑅,𝑦   𝑥,𝑋,𝑦

Proof of Theorem sup0
StepHypRef Expression
1 sup0riota 8995 . . 3 (𝑅 Or 𝐴 → sup(∅, 𝐴, 𝑅) = (𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥))
213ad2ant1 1134 . 2 ((𝑅 Or 𝐴 ∧ (𝑋𝐴 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑋) ∧ ∃!𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥) → sup(∅, 𝐴, 𝑅) = (𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥))
3 simp2r 1201 . . 3 ((𝑅 Or 𝐴 ∧ (𝑋𝐴 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑋) ∧ ∃!𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥) → ∀𝑦𝐴 ¬ 𝑦𝑅𝑋)
4 simpl 486 . . . . . 6 ((𝑋𝐴 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑋) → 𝑋𝐴)
54anim1i 618 . . . . 5 (((𝑋𝐴 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑋) ∧ ∃!𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥) → (𝑋𝐴 ∧ ∃!𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥))
653adant1 1131 . . . 4 ((𝑅 Or 𝐴 ∧ (𝑋𝐴 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑋) ∧ ∃!𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥) → (𝑋𝐴 ∧ ∃!𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥))
7 breq2 5031 . . . . . . 7 (𝑥 = 𝑋 → (𝑦𝑅𝑥𝑦𝑅𝑋))
87notbid 321 . . . . . 6 (𝑥 = 𝑋 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅𝑋))
98ralbidv 3109 . . . . 5 (𝑥 = 𝑋 → (∀𝑦𝐴 ¬ 𝑦𝑅𝑥 ↔ ∀𝑦𝐴 ¬ 𝑦𝑅𝑋))
109riota2 7147 . . . 4 ((𝑋𝐴 ∧ ∃!𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥) → (∀𝑦𝐴 ¬ 𝑦𝑅𝑋 ↔ (𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥) = 𝑋))
116, 10syl 17 . . 3 ((𝑅 Or 𝐴 ∧ (𝑋𝐴 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑋) ∧ ∃!𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥) → (∀𝑦𝐴 ¬ 𝑦𝑅𝑋 ↔ (𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥) = 𝑋))
123, 11mpbid 235 . 2 ((𝑅 Or 𝐴 ∧ (𝑋𝐴 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑋) ∧ ∃!𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥) → (𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥) = 𝑋)
132, 12eqtrd 2773 1 ((𝑅 Or 𝐴 ∧ (𝑋𝐴 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑋) ∧ ∃!𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥) → sup(∅, 𝐴, 𝑅) = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1088   = wceq 1542  wcel 2113  wral 3053  ∃!wreu 3055  c0 4209   class class class wbr 5027   Or wor 5437  crio 7120  supcsup 8970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3399  df-sbc 3680  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-sn 4514  df-pr 4516  df-op 4520  df-uni 4794  df-br 5028  df-po 5438  df-so 5439  df-iota 6291  df-riota 7121  df-sup 8972
This theorem is referenced by:  infempty  9037
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