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Theorem sup0 9461
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 9460 . . 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 484 . . . . . 6 ((𝑋𝐴 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑋) → 𝑋𝐴)
54anim1i 616 . . . . 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 5153 . . . . . . 7 (𝑥 = 𝑋 → (𝑦𝑅𝑥𝑦𝑅𝑋))
87notbid 318 . . . . . 6 (𝑥 = 𝑋 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅𝑋))
98ralbidv 3178 . . . . 5 (𝑥 = 𝑋 → (∀𝑦𝐴 ¬ 𝑦𝑅𝑥 ↔ ∀𝑦𝐴 ¬ 𝑦𝑅𝑋))
109riota2 7391 . . . 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 231 . 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 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wral 3062  ∃!wreu 3375  c0 4323   class class class wbr 5149   Or wor 5588  crio 7364  supcsup 9435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-po 5589  df-so 5590  df-iota 6496  df-riota 7365  df-sup 9437
This theorem is referenced by:  infempty  9502
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