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Theorem supgtoreq 9190
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 4274 . . . . . . . 8 (𝜑𝐵 ≠ ∅)
7 fisup2g 9188 . . . . . . . 8 ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵𝐴)) → ∃𝑥𝐵 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
83, 5, 6, 4, 7syl13anc 1370 . . . . . . 7 (𝜑 → ∃𝑥𝐵 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
9 ssrexv 3992 . . . . . . 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 9179 . . . . 5 (𝜑 → (𝐶𝐵 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))
122, 11mpd 15 . . . 4 (𝜑 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝐶)
131, 12eqnbrtrd 5096 . . 3 (𝜑 → ¬ 𝑆𝑅𝐶)
14 fisupcl 9189 . . . . . . . 8 ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵𝐴)) → sup(𝐵, 𝐴, 𝑅) ∈ 𝐵)
153, 5, 6, 4, 14syl13anc 1370 . . . . . . 7 (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐵)
164, 15sseldd 3926 . . . . . 6 (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)
171, 16eqeltrd 2840 . . . . 5 (𝜑𝑆𝐴)
184, 2sseldd 3926 . . . . 5 (𝜑𝐶𝐴)
19 sotric 5530 . . . . 5 ((𝑅 Or 𝐴 ∧ (𝑆𝐴𝐶𝐴)) → (𝑆𝑅𝐶 ↔ ¬ (𝑆 = 𝐶𝐶𝑅𝑆)))
203, 17, 18, 19syl12anc 833 . . . 4 (𝜑 → (𝑆𝑅𝐶 ↔ ¬ (𝑆 = 𝐶𝐶𝑅𝑆)))
21 orcom 866 . . . . . 6 ((𝑆 = 𝐶𝐶𝑅𝑆) ↔ (𝐶𝑅𝑆𝑆 = 𝐶))
22 eqcom 2746 . . . . . . 7 (𝑆 = 𝐶𝐶 = 𝑆)
2322orbi2i 909 . . . . . 6 ((𝐶𝑅𝑆𝑆 = 𝐶) ↔ (𝐶𝑅𝑆𝐶 = 𝑆))
2421, 23bitri 274 . . . . 5 ((𝑆 = 𝐶𝐶𝑅𝑆) ↔ (𝐶𝑅𝑆𝐶 = 𝑆))
2524notbii 319 . . . 4 (¬ (𝑆 = 𝐶𝐶𝑅𝑆) ↔ ¬ (𝐶𝑅𝑆𝐶 = 𝑆))
2620, 25bitr2di 287 . . 3 (𝜑 → (¬ (𝐶𝑅𝑆𝐶 = 𝑆) ↔ 𝑆𝑅𝐶))
2713, 26mtbird 324 . 2 (𝜑 → ¬ ¬ (𝐶𝑅𝑆𝐶 = 𝑆))
2827notnotrd 133 1 (𝜑 → (𝐶𝑅𝑆𝐶 = 𝑆))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  wo 843   = wceq 1541  wcel 2109  wne 2944  wral 3065  wrex 3066  wss 3891  c0 4261   class class class wbr 5078   Or wor 5501  Fincfn 8707  supcsup 9160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pr 5355  ax-un 7579
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-reu 3072  df-rmo 3073  df-rab 3074  df-v 3432  df-sbc 3720  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-pss 3910  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-tp 4571  df-op 4573  df-uni 4845  df-br 5079  df-opab 5141  df-tr 5196  df-id 5488  df-eprel 5494  df-po 5502  df-so 5503  df-fr 5543  df-we 5545  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-ord 6266  df-on 6267  df-lim 6268  df-suc 6269  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-riota 7225  df-om 7701  df-en 8708  df-fin 8711  df-sup 9162
This theorem is referenced by:  infltoreq  9222  supfirege  11945
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