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Theorem sup0 9225
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 9224 . . 3 (𝑅 Or 𝐴 → sup(∅, 𝐴, 𝑅) = (𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥))
213ad2ant1 1132 . 2 ((𝑅 Or 𝐴 ∧ (𝑋𝐴 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑋) ∧ ∃!𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥) → sup(∅, 𝐴, 𝑅) = (𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥))
3 simp2r 1199 . . 3 ((𝑅 Or 𝐴 ∧ (𝑋𝐴 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑋) ∧ ∃!𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥) → ∀𝑦𝐴 ¬ 𝑦𝑅𝑋)
4 simpl 483 . . . . . 6 ((𝑋𝐴 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑋) → 𝑋𝐴)
54anim1i 615 . . . . 5 (((𝑋𝐴 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑋) ∧ ∃!𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥) → (𝑋𝐴 ∧ ∃!𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥))
653adant1 1129 . . . 4 ((𝑅 Or 𝐴 ∧ (𝑋𝐴 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑋) ∧ ∃!𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥) → (𝑋𝐴 ∧ ∃!𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥))
7 breq2 5078 . . . . . . 7 (𝑥 = 𝑋 → (𝑦𝑅𝑥𝑦𝑅𝑋))
87notbid 318 . . . . . 6 (𝑥 = 𝑋 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅𝑋))
98ralbidv 3112 . . . . 5 (𝑥 = 𝑋 → (∀𝑦𝐴 ¬ 𝑦𝑅𝑥 ↔ ∀𝑦𝐴 ¬ 𝑦𝑅𝑋))
109riota2 7258 . . . 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 2778 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 396  w3a 1086   = wceq 1539  wcel 2106  wral 3064  ∃!wreu 3066  c0 4256   class class class wbr 5074   Or wor 5502  crio 7231  supcsup 9199
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-po 5503  df-so 5504  df-iota 6391  df-riota 7232  df-sup 9201
This theorem is referenced by:  infempty  9266
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