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Theorem suplocexprlemss 7716
Description: Lemma for suplocexpr 7726. 𝐴 is a set of positive reals. (Contributed by Jim Kingdon, 7-Jan-2024.)
Hypotheses
Ref Expression
suplocexpr.m (𝜑 → ∃𝑥 𝑥𝐴)
suplocexpr.ub (𝜑 → ∃𝑥P𝑦𝐴 𝑦<P 𝑥)
suplocexpr.loc (𝜑 → ∀𝑥P𝑦P (𝑥<P 𝑦 → (∃𝑧𝐴 𝑥<P 𝑧 ∨ ∀𝑧𝐴 𝑧<P 𝑦)))
Assertion
Ref Expression
suplocexprlemss (𝜑𝐴P)
Distinct variable groups:   𝑥,𝐴,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑧)   𝐴(𝑧)

Proof of Theorem suplocexprlemss
StepHypRef Expression
1 suplocexpr.ub . . 3 (𝜑 → ∃𝑥P𝑦𝐴 𝑦<P 𝑥)
2 rsp 2524 . . . . . 6 (∀𝑦𝐴 𝑦<P 𝑥 → (𝑦𝐴𝑦<P 𝑥))
3 ltrelpr 7506 . . . . . . . 8 <P ⊆ (P × P)
43brel 4680 . . . . . . 7 (𝑦<P 𝑥 → (𝑦P𝑥P))
54simpld 112 . . . . . 6 (𝑦<P 𝑥𝑦P)
62, 5syl6 33 . . . . 5 (∀𝑦𝐴 𝑦<P 𝑥 → (𝑦𝐴𝑦P))
76a1i 9 . . . 4 (𝜑 → (∀𝑦𝐴 𝑦<P 𝑥 → (𝑦𝐴𝑦P)))
87rexlimdvw 2598 . . 3 (𝜑 → (∃𝑥P𝑦𝐴 𝑦<P 𝑥 → (𝑦𝐴𝑦P)))
91, 8mpd 13 . 2 (𝜑 → (𝑦𝐴𝑦P))
109ssrdv 3163 1 (𝜑𝐴P)
Colors of variables: wff set class
Syntax hints:  wi 4  wo 708  wex 1492  wcel 2148  wral 2455  wrex 2456  wss 3131   class class class wbr 4005  Pcnp 7292  <P cltp 7296
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-opab 4067  df-xp 4634  df-iltp 7471
This theorem is referenced by:  suplocexprlemml  7717  suplocexprlemrl  7718  suplocexprlemmu  7719  suplocexprlemru  7720  suplocexprlemdisj  7721  suplocexprlemloc  7722  suplocexprlemex  7723  suplocexprlemub  7724  suplocexprlemlub  7725
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