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Theorem supgtoreq 8417
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.4 . . . . 5 (𝜑𝐶𝐵)
2 supgtoreq.1 . . . . . 6 (𝜑𝑅 Or 𝐴)
3 supgtoreq.2 . . . . . . 7 (𝜑𝐵𝐴)
4 supgtoreq.3 . . . . . . . 8 (𝜑𝐵 ∈ Fin)
5 ne0i 3954 . . . . . . . . 9 (𝐶𝐵𝐵 ≠ ∅)
61, 5syl 17 . . . . . . . 8 (𝜑𝐵 ≠ ∅)
7 fisup2g 8415 . . . . . . . 8 ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵𝐴)) → ∃𝑥𝐵 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
82, 4, 6, 3, 7syl13anc 1368 . . . . . . 7 (𝜑 → ∃𝑥𝐵 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
9 ssrexv 3700 . . . . . . 7 (𝐵𝐴 → (∃𝑥𝐵 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)) → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧))))
103, 8, 9sylc 65 . . . . . 6 (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
112, 10supub 8406 . . . . 5 (𝜑 → (𝐶𝐵 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))
121, 11mpd 15 . . . 4 (𝜑 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝐶)
13 supgtoreq.5 . . . . 5 (𝜑𝑆 = sup(𝐵, 𝐴, 𝑅))
1413breq1d 4695 . . . 4 (𝜑 → (𝑆𝑅𝐶 ↔ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))
1512, 14mtbird 314 . . 3 (𝜑 → ¬ 𝑆𝑅𝐶)
16 fisupcl 8416 . . . . . . . 8 ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵𝐴)) → sup(𝐵, 𝐴, 𝑅) ∈ 𝐵)
172, 4, 6, 3, 16syl13anc 1368 . . . . . . 7 (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐵)
183, 17sseldd 3637 . . . . . 6 (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)
1913, 18eqeltrd 2730 . . . . 5 (𝜑𝑆𝐴)
203, 1sseldd 3637 . . . . 5 (𝜑𝐶𝐴)
21 sotric 5090 . . . . 5 ((𝑅 Or 𝐴 ∧ (𝑆𝐴𝐶𝐴)) → (𝑆𝑅𝐶 ↔ ¬ (𝑆 = 𝐶𝐶𝑅𝑆)))
222, 19, 20, 21syl12anc 1364 . . . 4 (𝜑 → (𝑆𝑅𝐶 ↔ ¬ (𝑆 = 𝐶𝐶𝑅𝑆)))
23 orcom 401 . . . . . 6 ((𝑆 = 𝐶𝐶𝑅𝑆) ↔ (𝐶𝑅𝑆𝑆 = 𝐶))
24 eqcom 2658 . . . . . . 7 (𝑆 = 𝐶𝐶 = 𝑆)
2524orbi2i 540 . . . . . 6 ((𝐶𝑅𝑆𝑆 = 𝐶) ↔ (𝐶𝑅𝑆𝐶 = 𝑆))
2623, 25bitri 264 . . . . 5 ((𝑆 = 𝐶𝐶𝑅𝑆) ↔ (𝐶𝑅𝑆𝐶 = 𝑆))
2726notbii 309 . . . 4 (¬ (𝑆 = 𝐶𝐶𝑅𝑆) ↔ ¬ (𝐶𝑅𝑆𝐶 = 𝑆))
2822, 27syl6rbb 277 . . 3 (𝜑 → (¬ (𝐶𝑅𝑆𝐶 = 𝑆) ↔ 𝑆𝑅𝐶))
2915, 28mtbird 314 . 2 (𝜑 → ¬ ¬ (𝐶𝑅𝑆𝐶 = 𝑆))
3029notnotrd 128 1 (𝜑 → (𝐶𝑅𝑆𝐶 = 𝑆))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383   = wceq 1523  wcel 2030  wne 2823  wral 2941  wrex 2942  wss 3607  c0 3948   class class class wbr 4685   Or wor 5063  Fincfn 7997  supcsup 8387
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-8 2032  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-pow 4873  ax-pr 4936  ax-un 6991
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-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-om 7108  df-1o 7605  df-er 7787  df-en 7998  df-fin 8001  df-sup 8389
This theorem is referenced by:  infltoreq  8449  supfirege  11047
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