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Theorem supgtoreq 8361
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 3913 . . . . . . . . 9 (𝐶𝐵𝐵 ≠ ∅)
61, 5syl 17 . . . . . . . 8 (𝜑𝐵 ≠ ∅)
7 fisup2g 8359 . . . . . . . 8 ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵𝐴)) → ∃𝑥𝐵 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
82, 4, 6, 3, 7syl13anc 1326 . . . . . . 7 (𝜑 → ∃𝑥𝐵 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))
9 ssrexv 3659 . . . . . . 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 8350 . . . . 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 4654 . . . 4 (𝜑 → (𝑆𝑅𝐶 ↔ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))
1512, 14mtbird 315 . . 3 (𝜑 → ¬ 𝑆𝑅𝐶)
16 fisupcl 8360 . . . . . . . 8 ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵𝐴)) → sup(𝐵, 𝐴, 𝑅) ∈ 𝐵)
172, 4, 6, 3, 16syl13anc 1326 . . . . . . 7 (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐵)
183, 17sseldd 3596 . . . . . 6 (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)
1913, 18eqeltrd 2699 . . . . 5 (𝜑𝑆𝐴)
203, 1sseldd 3596 . . . . 5 (𝜑𝐶𝐴)
21 sotric 5051 . . . . 5 ((𝑅 Or 𝐴 ∧ (𝑆𝐴𝐶𝐴)) → (𝑆𝑅𝐶 ↔ ¬ (𝑆 = 𝐶𝐶𝑅𝑆)))
222, 19, 20, 21syl12anc 1322 . . . 4 (𝜑 → (𝑆𝑅𝐶 ↔ ¬ (𝑆 = 𝐶𝐶𝑅𝑆)))
23 orcom 402 . . . . . 6 ((𝑆 = 𝐶𝐶𝑅𝑆) ↔ (𝐶𝑅𝑆𝑆 = 𝐶))
24 eqcom 2627 . . . . . . 7 (𝑆 = 𝐶𝐶 = 𝑆)
2524orbi2i 541 . . . . . 6 ((𝐶𝑅𝑆𝑆 = 𝐶) ↔ (𝐶𝑅𝑆𝐶 = 𝑆))
2623, 25bitri 264 . . . . 5 ((𝑆 = 𝐶𝐶𝑅𝑆) ↔ (𝐶𝑅𝑆𝐶 = 𝑆))
2726notbii 310 . . . 4 (¬ (𝑆 = 𝐶𝐶𝑅𝑆) ↔ ¬ (𝐶𝑅𝑆𝐶 = 𝑆))
2822, 27syl6rbb 277 . . 3 (𝜑 → (¬ (𝐶𝑅𝑆𝐶 = 𝑆) ↔ 𝑆𝑅𝐶))
2915, 28mtbird 315 . 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 383  wa 384   = wceq 1481  wcel 1988  wne 2791  wral 2909  wrex 2910  wss 3567  c0 3907   class class class wbr 4644   Or wor 5024  Fincfn 7940  supcsup 8331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-om 7051  df-1o 7545  df-er 7727  df-en 7941  df-fin 7944  df-sup 8333
This theorem is referenced by:  infltoreq  8393  supfirege  10994
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