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Theorem supgtoreq 8931
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 4284 . . . . . . . 8 (𝜑𝐵 ≠ ∅)
7 fisup2g 8929 . . . . . . . 8 ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵𝐴)) → ∃𝑥𝐵 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
83, 5, 6, 4, 7syl13anc 1369 . . . . . . 7 (𝜑 → ∃𝑥𝐵 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
9 ssrexv 4020 . . . . . . 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 8920 . . . . 5 (𝜑 → (𝐶𝐵 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))
122, 11mpd 15 . . . 4 (𝜑 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝐶)
131, 12eqnbrtrd 5070 . . 3 (𝜑 → ¬ 𝑆𝑅𝐶)
14 fisupcl 8930 . . . . . . . 8 ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵𝐴)) → sup(𝐵, 𝐴, 𝑅) ∈ 𝐵)
153, 5, 6, 4, 14syl13anc 1369 . . . . . . 7 (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐵)
164, 15sseldd 3954 . . . . . 6 (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)
171, 16eqeltrd 2916 . . . . 5 (𝜑𝑆𝐴)
184, 2sseldd 3954 . . . . 5 (𝜑𝐶𝐴)
19 sotric 5488 . . . . 5 ((𝑅 Or 𝐴 ∧ (𝑆𝐴𝐶𝐴)) → (𝑆𝑅𝐶 ↔ ¬ (𝑆 = 𝐶𝐶𝑅𝑆)))
203, 17, 18, 19syl12anc 835 . . . 4 (𝜑 → (𝑆𝑅𝐶 ↔ ¬ (𝑆 = 𝐶𝐶𝑅𝑆)))
21 orcom 867 . . . . . 6 ((𝑆 = 𝐶𝐶𝑅𝑆) ↔ (𝐶𝑅𝑆𝑆 = 𝐶))
22 eqcom 2831 . . . . . . 7 (𝑆 = 𝐶𝐶 = 𝑆)
2322orbi2i 910 . . . . . 6 ((𝐶𝑅𝑆𝑆 = 𝐶) ↔ (𝐶𝑅𝑆𝐶 = 𝑆))
2421, 23bitri 278 . . . . 5 ((𝑆 = 𝐶𝐶𝑅𝑆) ↔ (𝐶𝑅𝑆𝐶 = 𝑆))
2524notbii 323 . . . 4 (¬ (𝑆 = 𝐶𝐶𝑅𝑆) ↔ ¬ (𝐶𝑅𝑆𝐶 = 𝑆))
2620, 25syl6rbb 291 . . 3 (𝜑 → (¬ (𝐶𝑅𝑆𝐶 = 𝑆) ↔ 𝑆𝑅𝐶))
2713, 26mtbird 328 . 2 (𝜑 → ¬ ¬ (𝐶𝑅𝑆𝐶 = 𝑆))
2827notnotrd 135 1 (𝜑 → (𝐶𝑅𝑆𝐶 = 𝑆))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844   = wceq 1538  wcel 2115  wne 3014  wral 3133  wrex 3134  wss 3919  c0 4276   class class class wbr 5052   Or wor 5460  Fincfn 8505  supcsup 8901
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rmo 3141  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-om 7575  df-1o 8098  df-er 8285  df-en 8506  df-fin 8509  df-sup 8903
This theorem is referenced by:  infltoreq  8963  supfirege  11623
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