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Theorem eqinfd 8937
Description: Sufficient condition for an element to be equal to the infimum. (Contributed by AV, 3-Sep-2020.)
Hypotheses
Ref Expression
infexd.1 (𝜑𝑅 Or 𝐴)
eqinfd.2 (𝜑𝐶𝐴)
eqinfd.3 ((𝜑𝑦𝐵) → ¬ 𝑦𝑅𝐶)
eqinfd.4 ((𝜑 ∧ (𝑦𝐴𝐶𝑅𝑦)) → ∃𝑧𝐵 𝑧𝑅𝑦)
Assertion
Ref Expression
eqinfd (𝜑 → inf(𝐵, 𝐴, 𝑅) = 𝐶)
Distinct variable groups:   𝑦,𝐴,𝑧   𝑦,𝐵,𝑧   𝑦,𝑅,𝑧   𝑦,𝐶,𝑧   𝜑,𝑦
Allowed substitution hint:   𝜑(𝑧)

Proof of Theorem eqinfd
StepHypRef Expression
1 eqinfd.2 . 2 (𝜑𝐶𝐴)
2 eqinfd.3 . . 3 ((𝜑𝑦𝐵) → ¬ 𝑦𝑅𝐶)
32ralrimiva 3152 . 2 (𝜑 → ∀𝑦𝐵 ¬ 𝑦𝑅𝐶)
4 eqinfd.4 . . . 4 ((𝜑 ∧ (𝑦𝐴𝐶𝑅𝑦)) → ∃𝑧𝐵 𝑧𝑅𝑦)
54expr 460 . . 3 ((𝜑𝑦𝐴) → (𝐶𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦))
65ralrimiva 3152 . 2 (𝜑 → ∀𝑦𝐴 (𝐶𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦))
7 infexd.1 . . 3 (𝜑𝑅 Or 𝐴)
87eqinf 8936 . 2 (𝜑 → ((𝐶𝐴 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝐶 ∧ ∀𝑦𝐴 (𝐶𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)) → inf(𝐵, 𝐴, 𝑅) = 𝐶))
91, 3, 6, 8mp3and 1461 1 (𝜑 → inf(𝐵, 𝐴, 𝑅) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399   = wceq 1538  wcel 2112  wral 3109  wrex 3110   class class class wbr 5033   Or wor 5441  infcinf 8893
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rmo 3117  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-po 5442  df-so 5443  df-cnv 5531  df-iota 6287  df-riota 7097  df-sup 8894  df-inf 8895
This theorem is referenced by:  infmin  8946  xrinf0  12723  infmremnf  12728  infmrp1  12729
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