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Theorem eqinfd 8274
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 2949 . 2 (𝜑 → ∀𝑦𝐵 ¬ 𝑦𝑅𝐶)
4 eqinfd.4 . . . 4 ((𝜑 ∧ (𝑦𝐴𝐶𝑅𝑦)) → ∃𝑧𝐵 𝑧𝑅𝑦)
54expr 641 . . 3 ((𝜑𝑦𝐴) → (𝐶𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦))
65ralrimiva 2949 . 2 (𝜑 → ∀𝑦𝐴 (𝐶𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦))
7 infexd.1 . . 3 (𝜑𝑅 Or 𝐴)
87eqinf 8273 . 2 (𝜑 → ((𝐶𝐴 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝐶 ∧ ∀𝑦𝐴 (𝐶𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)) → inf(𝐵, 𝐴, 𝑅) = 𝐶))
91, 3, 6, 8mp3and 1419 1 (𝜑 → inf(𝐵, 𝐴, 𝑅) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383   = wceq 1475  wcel 1977  wral 2896  wrex 2897   class class class wbr 4583   Or wor 4958  infcinf 8230
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-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
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-opab 4644  df-po 4959  df-so 4960  df-cnv 5046  df-iota 5768  df-riota 6511  df-sup 8231  df-inf 8232
This theorem is referenced by:  infmin  8283  xrinf0  12039  infmremnf  12044  infmrp1  12045
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