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Theorem eqinfd 8666
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 3175 . 2 (𝜑 → ∀𝑦𝐵 ¬ 𝑦𝑅𝐶)
4 eqinfd.4 . . . 4 ((𝜑 ∧ (𝑦𝐴𝐶𝑅𝑦)) → ∃𝑧𝐵 𝑧𝑅𝑦)
54expr 450 . . 3 ((𝜑𝑦𝐴) → (𝐶𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦))
65ralrimiva 3175 . 2 (𝜑 → ∀𝑦𝐴 (𝐶𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦))
7 infexd.1 . . 3 (𝜑𝑅 Or 𝐴)
87eqinf 8665 . 2 (𝜑 → ((𝐶𝐴 ∧ ∀𝑦𝐵 ¬ 𝑦𝑅𝐶 ∧ ∀𝑦𝐴 (𝐶𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)) → inf(𝐵, 𝐴, 𝑅) = 𝐶))
91, 3, 6, 8mp3and 1592 1 (𝜑 → inf(𝐵, 𝐴, 𝑅) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 386   = wceq 1656  wcel 2164  wral 3117  wrex 3118   class class class wbr 4875   Or wor 5264  infcinf 8622
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-po 5265  df-so 5266  df-cnv 5354  df-iota 6090  df-riota 6871  df-sup 8623  df-inf 8624
This theorem is referenced by:  infmin  8675  xrinf0  12463  infmremnf  12468  infmrp1  12469
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