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Theorem supgtoreq 9508
Description: The supremum of a finite set is greater than or equal to all the elements of the set. (Contributed by AV, 1-Oct-2019.)
Hypotheses
Ref Expression
supgtoreq.1 (𝜑𝑅 Or 𝐴)
supgtoreq.2 (𝜑𝐵𝐴)
supgtoreq.3 (𝜑𝐵 ∈ Fin)
supgtoreq.4 (𝜑𝐶𝐵)
supgtoreq.5 (𝜑𝑆 = sup(𝐵, 𝐴, 𝑅))
Assertion
Ref Expression
supgtoreq (𝜑 → (𝐶𝑅𝑆𝐶 = 𝑆))

Proof of Theorem supgtoreq
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 supgtoreq.5 . . . 4 (𝜑𝑆 = sup(𝐵, 𝐴, 𝑅))
2 supgtoreq.4 . . . . 5 (𝜑𝐶𝐵)
3 supgtoreq.1 . . . . . 6 (𝜑𝑅 Or 𝐴)
4 supgtoreq.2 . . . . . . 7 (𝜑𝐵𝐴)
5 supgtoreq.3 . . . . . . . 8 (𝜑𝐵 ∈ Fin)
62ne0d 4348 . . . . . . . 8 (𝜑𝐵 ≠ ∅)
7 fisup2g 9506 . . . . . . . 8 ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵𝐴)) → ∃𝑥𝐵 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
83, 5, 6, 4, 7syl13anc 1371 . . . . . . 7 (𝜑 → ∃𝑥𝐵 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
9 ssrexv 4065 . . . . . . 7 (𝐵𝐴 → (∃𝑥𝐵 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)) → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))))
104, 8, 9sylc 65 . . . . . 6 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
113, 10supub 9497 . . . . 5 (𝜑 → (𝐶𝐵 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))
122, 11mpd 15 . . . 4 (𝜑 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝐶)
131, 12eqnbrtrd 5166 . . 3 (𝜑 → ¬ 𝑆𝑅𝐶)
14 fisupcl 9507 . . . . . . . 8 ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵𝐴)) → sup(𝐵, 𝐴, 𝑅) ∈ 𝐵)
153, 5, 6, 4, 14syl13anc 1371 . . . . . . 7 (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐵)
164, 15sseldd 3996 . . . . . 6 (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)
171, 16eqeltrd 2839 . . . . 5 (𝜑𝑆𝐴)
184, 2sseldd 3996 . . . . 5 (𝜑𝐶𝐴)
19 sotric 5626 . . . . 5 ((𝑅 Or 𝐴 ∧ (𝑆𝐴𝐶𝐴)) → (𝑆𝑅𝐶 ↔ ¬ (𝑆 = 𝐶𝐶𝑅𝑆)))
203, 17, 18, 19syl12anc 837 . . . 4 (𝜑 → (𝑆𝑅𝐶 ↔ ¬ (𝑆 = 𝐶𝐶𝑅𝑆)))
21 orcom 870 . . . . . 6 ((𝑆 = 𝐶𝐶𝑅𝑆) ↔ (𝐶𝑅𝑆𝑆 = 𝐶))
22 eqcom 2742 . . . . . . 7 (𝑆 = 𝐶𝐶 = 𝑆)
2322orbi2i 912 . . . . . 6 ((𝐶𝑅𝑆𝑆 = 𝐶) ↔ (𝐶𝑅𝑆𝐶 = 𝑆))
2421, 23bitri 275 . . . . 5 ((𝑆 = 𝐶𝐶𝑅𝑆) ↔ (𝐶𝑅𝑆𝐶 = 𝑆))
2524notbii 320 . . . 4 (¬ (𝑆 = 𝐶𝐶𝑅𝑆) ↔ ¬ (𝐶𝑅𝑆𝐶 = 𝑆))
2620, 25bitr2di 288 . . 3 (𝜑 → (¬ (𝐶𝑅𝑆𝐶 = 𝑆) ↔ 𝑆𝑅𝐶))
2713, 26mtbird 325 . 2 (𝜑 → ¬ ¬ (𝐶𝑅𝑆𝐶 = 𝑆))
2827notnotrd 133 1 (𝜑 → (𝐶𝑅𝑆𝐶 = 𝑆))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wo 847   = wceq 1537  wcel 2106  wne 2938  wral 3059  wrex 3068  wss 3963  c0 4339   class class class wbr 5148   Or wor 5596  Fincfn 8984  supcsup 9478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-om 7888  df-en 8985  df-fin 8988  df-sup 9480
This theorem is referenced by:  infltoreq  9540  supfirege  12253  safesnsupfilb  43408
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