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Theorem sup0 8918
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 8917 . . 3 (𝑅 Or 𝐴 → sup(∅, 𝐴, 𝑅) = (𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥))
213ad2ant1 1125 . 2 ((𝑅 Or 𝐴 ∧ (𝑋𝐴 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑋) ∧ ∃!𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥) → sup(∅, 𝐴, 𝑅) = (𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥))
3 simp2r 1192 . . 3 ((𝑅 Or 𝐴 ∧ (𝑋𝐴 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑋) ∧ ∃!𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥) → ∀𝑦𝐴 ¬ 𝑦𝑅𝑋)
4 simpl 483 . . . . . 6 ((𝑋𝐴 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑋) → 𝑋𝐴)
54anim1i 614 . . . . 5 (((𝑋𝐴 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑋) ∧ ∃!𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥) → (𝑋𝐴 ∧ ∃!𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥))
653adant1 1122 . . . 4 ((𝑅 Or 𝐴 ∧ (𝑋𝐴 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑋) ∧ ∃!𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥) → (𝑋𝐴 ∧ ∃!𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥))
7 breq2 5061 . . . . . . 7 (𝑥 = 𝑋 → (𝑦𝑅𝑥𝑦𝑅𝑋))
87notbid 319 . . . . . 6 (𝑥 = 𝑋 → (¬ 𝑦𝑅𝑥 ↔ ¬ 𝑦𝑅𝑋))
98ralbidv 3194 . . . . 5 (𝑥 = 𝑋 → (∀𝑦𝐴 ¬ 𝑦𝑅𝑥 ↔ ∀𝑦𝐴 ¬ 𝑦𝑅𝑋))
109riota2 7128 . . . 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 233 . 2 ((𝑅 Or 𝐴 ∧ (𝑋𝐴 ∧ ∀𝑦𝐴 ¬ 𝑦𝑅𝑋) ∧ ∃!𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥) → (𝑥𝐴𝑦𝐴 ¬ 𝑦𝑅𝑥) = 𝑋)
132, 12eqtrd 2853 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 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  ∃!wreu 3137  c0 4288   class class class wbr 5057   Or wor 5466  crio 7102  supcsup 8892
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-po 5467  df-so 5468  df-iota 6307  df-riota 7103  df-sup 8894
This theorem is referenced by:  infempty  8959
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