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Theorem eqinfd 9510
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 3135 . 2 (𝜑 → ∀𝑦𝐵 ¬ 𝑦𝑅𝐶)
4 eqinfd.4 . . . 4 ((𝜑 ∧ (𝑦𝐴𝐶𝑅𝑦)) → ∃𝑧𝐵 𝑧𝑅𝑦)
54expr 455 . . 3 ((𝜑𝑦𝐴) → (𝐶𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦))
65ralrimiva 3135 . 2 (𝜑 → ∀𝑦𝐴 (𝐶𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦))
7 infexd.1 . . 3 (𝜑𝑅 Or 𝐴)
87eqinf 9509 . 2 (𝜑 → ((𝐶𝐴 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝐶 ∧ ∀𝑦𝐴 (𝐶𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)) → inf(𝐵, 𝐴, 𝑅) = 𝐶))
91, 3, 6, 8mp3and 1460 1 (𝜑 → inf(𝐵, 𝐴, 𝑅) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 394   = wceq 1533  wcel 2098  wral 3050  wrex 3059   class class class wbr 5149   Or wor 5589  infcinf 9466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-po 5590  df-so 5591  df-cnv 5686  df-iota 6501  df-riota 7375  df-sup 9467  df-inf 9468
This theorem is referenced by:  infmin  9519  xrinf0  13352  infmremnf  13357  infmrp1  13358
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