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Theorem infssd 29828
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 8554 . 2 (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ 𝐴)
4 infssd.1 . . . . 5 (𝜑𝐶𝐵)
54sseld 3751 . . . 4 (𝜑 → (𝑧𝐶𝑧𝐵))
61, 2inflb 8555 . . . 4 (𝜑 → (𝑧𝐵 → ¬ 𝑧𝑅inf(𝐵, 𝐴, 𝑅)))
75, 6syld 47 . . 3 (𝜑 → (𝑧𝐶 → ¬ 𝑧𝑅inf(𝐵, 𝐴, 𝑅)))
87ralrimiv 3114 . 2 (𝜑 → ∀𝑧𝐶 ¬ 𝑧𝑅inf(𝐵, 𝐴, 𝑅))
9 infssd.3 . . 3 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐶 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐶 𝑧𝑅𝑦)))
101, 9infnlb 8558 . 2 (𝜑 → ((inf(𝐵, 𝐴, 𝑅) ∈ 𝐴 ∧ ∀𝑧𝐶 ¬ 𝑧𝑅inf(𝐵, 𝐴, 𝑅)) → ¬ inf(𝐶, 𝐴, 𝑅)𝑅inf(𝐵, 𝐴, 𝑅)))
113, 8, 10mp2and 679 1 (𝜑 → ¬ inf(𝐶, 𝐴, 𝑅)𝑅inf(𝐵, 𝐴, 𝑅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  wcel 2145  wral 3061  wrex 3062  wss 3723   class class class wbr 4787   Or wor 5170  infcinf 8507
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-po 5171  df-so 5172  df-cnv 5258  df-iota 5993  df-riota 6757  df-sup 8508  df-inf 8509
This theorem is referenced by:  xrge0infssd  29866
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