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Theorem sup0 8255
 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 8254 . . 3 (𝑅 Or 𝐴 → sup(∅, 𝐴, 𝑅) = (𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥))
213ad2ant1 1075 . 2 ((𝑅 Or 𝐴 ∧ (𝑋𝐴 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑋) ∧ ∃!𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥) → sup(∅, 𝐴, 𝑅) = (𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥))
3 simp2r 1081 . . 3 ((𝑅 Or 𝐴 ∧ (𝑋𝐴 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑋) ∧ ∃!𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥) → ∀𝑦𝐴 ¬ 𝑦𝑅𝑋)
4 simpl 472 . . . . . 6 ((𝑋𝐴 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑋) → 𝑋𝐴)
54anim1i 590 . . . . 5 (((𝑋𝐴 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑋) ∧ ∃!𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥) → (𝑋𝐴 ∧ ∃!𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥))
653adant1 1072 . . . 4 ((𝑅 Or 𝐴 ∧ (𝑋𝐴 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑋) ∧ ∃!𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥) → (𝑋𝐴 ∧ ∃!𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥))
7 breq2 4587 . . . . . . 7 (𝑥 = 𝑋 → (𝑦𝑅𝑥𝑦𝑅𝑋))
87notbid 307 . . . . . 6 (𝑥 = 𝑋 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅𝑋))
98ralbidv 2969 . . . . 5 (𝑥 = 𝑋 → (∀𝑦𝐴 ¬ 𝑦𝑅𝑥 ↔ ∀𝑦𝐴 ¬ 𝑦𝑅𝑋))
109riota2 6533 . . . 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 221 . 2 ((𝑅 Or 𝐴 ∧ (𝑋𝐴 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑋) ∧ ∃!𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥) → (𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥) = 𝑋)
132, 12eqtrd 2644 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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃!wreu 2898  ∅c0 3874   class class class wbr 4583   Or wor 4958  ℩crio 6510  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:  infempty  8295
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