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Theorem infssd 30372
Description: Inequality deduction for infimum of a subset. (Contributed by AV, 4-Oct-2020.)
Hypotheses
Ref Expression
infssd.0 (𝜑𝑅 Or 𝐴)
infssd.1 (𝜑𝐶𝐵)
infssd.3 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐶 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐶 𝑧𝑅𝑦)))
infssd.4 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))
Assertion
Ref Expression
infssd (𝜑 → ¬ inf(𝐶, 𝐴, 𝑅)𝑅inf(𝐵, 𝐴, 𝑅))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧   𝜑,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem infssd
StepHypRef Expression
1 infssd.0 . . 3 (𝜑𝑅 Or 𝐴)
2 infssd.4 . . 3 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))
31, 2infcl 8940 . 2 (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ 𝐴)
4 infssd.1 . . . . 5 (𝜑𝐶𝐵)
54sseld 3963 . . . 4 (𝜑 → (𝑧𝐶𝑧𝐵))
61, 2inflb 8941 . . . 4 (𝜑 → (𝑧𝐵 → ¬ 𝑧𝑅inf(𝐵, 𝐴, 𝑅)))
75, 6syld 47 . . 3 (𝜑 → (𝑧𝐶 → ¬ 𝑧𝑅inf(𝐵, 𝐴, 𝑅)))
87ralrimiv 3178 . 2 (𝜑 → ∀𝑧𝐶 ¬ 𝑧𝑅inf(𝐵, 𝐴, 𝑅))
9 infssd.3 . . 3 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐶 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐶 𝑧𝑅𝑦)))
101, 9infnlb 8944 . 2 (𝜑 → ((inf(𝐵, 𝐴, 𝑅) ∈ 𝐴 ∧ ∀𝑧𝐶 ¬ 𝑧𝑅inf(𝐵, 𝐴, 𝑅)) → ¬ inf(𝐶, 𝐴, 𝑅)𝑅inf(𝐵, 𝐴, 𝑅)))
113, 8, 10mp2and 695 1 (𝜑 → ¬ inf(𝐶, 𝐴, 𝑅)𝑅inf(𝐵, 𝐴, 𝑅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wcel 2105  wral 3135  wrex 3136  wss 3933   class class class wbr 5057   Or wor 5466  infcinf 8893
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  ax-sep 5194  ax-nul 5201  ax-pr 5320
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-opab 5120  df-po 5467  df-so 5468  df-cnv 5556  df-iota 6307  df-riota 7103  df-sup 8894  df-inf 8895
This theorem is referenced by:  xrge0infssd  30411
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