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Theorem infssd 29616
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 8435 . 2 (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ 𝐴)
4 infssd.1 . . . . 5 (𝜑𝐶𝐵)
54sseld 3635 . . . 4 (𝜑 → (𝑧𝐶𝑧𝐵))
61, 2inflb 8436 . . . 4 (𝜑 → (𝑧𝐵 → ¬ 𝑧𝑅inf(𝐵, 𝐴, 𝑅)))
75, 6syld 47 . . 3 (𝜑 → (𝑧𝐶 → ¬ 𝑧𝑅inf(𝐵, 𝐴, 𝑅)))
87ralrimiv 2994 . 2 (𝜑 → ∀𝑧𝐶 ¬ 𝑧𝑅inf(𝐵, 𝐴, 𝑅))
9 infssd.3 . . 3 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐶 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐶 𝑧𝑅𝑦)))
101, 9infnlb 8439 . 2 (𝜑 → ((inf(𝐵, 𝐴, 𝑅) ∈ 𝐴 ∧ ∀𝑧𝐶 ¬ 𝑧𝑅inf(𝐵, 𝐴, 𝑅)) → ¬ inf(𝐶, 𝐴, 𝑅)𝑅inf(𝐵, 𝐴, 𝑅)))
113, 8, 10mp2and 715 1 (𝜑 → ¬ inf(𝐶, 𝐴, 𝑅)𝑅inf(𝐵, 𝐴, 𝑅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  wcel 2030  wral 2941  wrex 2942  wss 3607   class class class wbr 4685   Or wor 5063  infcinf 8388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-po 5064  df-so 5065  df-cnv 5151  df-iota 5889  df-riota 6651  df-sup 8389  df-inf 8390
This theorem is referenced by:  xrge0infssd  29654
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