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	⊢ (𝜑 → 𝑅 Or 𝐴)
supgtoreq.2	⊢ (𝜑 → 𝐵 ⊆ 𝐴)
supgtoreq.3	⊢ (𝜑 → 𝐵 ∈ Fin)
supgtoreq.4	⊢ (𝜑 → 𝐶 ∈ 𝐵)
supgtoreq.5	⊢ (𝜑 → 𝑆 = sup(𝐵, 𝐴, 𝑅))

Assertion

Ref	Expression
supgtoreq	⊢ (𝜑 → (𝐶𝑅𝑆 ∨ 𝐶 = 𝑆))

Proof of Theorem supgtoreq

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		supgtoreq.5	. . . 4 ⊢ (𝜑 → 𝑆 = sup(𝐵, 𝐴, 𝑅))
2		supgtoreq.4	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝐵)
3		supgtoreq.1	. . . . . 6 ⊢ (𝜑 → 𝑅 Or 𝐴)
4		supgtoreq.2	. . . . . . 7 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
5		supgtoreq.3	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ∈ Fin)
6	2	ne0d 4274	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ≠ ∅)
7		fisup2g 9188	. . . . . . . 8 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵 ⊆ 𝐴)) → ∃𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))
8	3, 5, 6, 4, 7	syl13anc 1370	. . . . . . 7 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))
9		ssrexv 3992	. . . . . . 7 ⊢ (𝐵 ⊆ 𝐴 → (∃𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))))
10	4, 8, 9	sylc 65	. . . . . 6 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))
11	3, 10	supub 9179	. . . . 5 ⊢ (𝜑 → (𝐶 ∈ 𝐵 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝐶))
12	2, 11	mpd 15	. . . 4 ⊢ (𝜑 → ¬ sup(𝐵, 𝐴, 𝑅)𝑅𝐶)
13	1, 12	eqnbrtrd 5096	. . 3 ⊢ (𝜑 → ¬ 𝑆𝑅𝐶)
14		fisupcl 9189	. . . . . . . 8 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵 ⊆ 𝐴)) → sup(𝐵, 𝐴, 𝑅) ∈ 𝐵)
15	3, 5, 6, 4, 14	syl13anc 1370	. . . . . . 7 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐵)
16	4, 15	sseldd 3926	. . . . . 6 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)
17	1, 16	eqeltrd 2840	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ 𝐴)
18	4, 2	sseldd 3926	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
19		sotric 5530	. . . . 5 ⊢ ((𝑅 Or 𝐴 ∧ (𝑆 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝑆𝑅𝐶 ↔ ¬ (𝑆 = 𝐶 ∨ 𝐶𝑅𝑆)))
20	3, 17, 18, 19	syl12anc 833	. . . 4 ⊢ (𝜑 → (𝑆𝑅𝐶 ↔ ¬ (𝑆 = 𝐶 ∨ 𝐶𝑅𝑆)))
21		orcom 866	. . . . . 6 ⊢ ((𝑆 = 𝐶 ∨ 𝐶𝑅𝑆) ↔ (𝐶𝑅𝑆 ∨ 𝑆 = 𝐶))
22		eqcom 2746	. . . . . . 7 ⊢ (𝑆 = 𝐶 ↔ 𝐶 = 𝑆)
23	22	orbi2i 909	. . . . . 6 ⊢ ((𝐶𝑅𝑆 ∨ 𝑆 = 𝐶) ↔ (𝐶𝑅𝑆 ∨ 𝐶 = 𝑆))
24	21, 23	bitri 274	. . . . 5 ⊢ ((𝑆 = 𝐶 ∨ 𝐶𝑅𝑆) ↔ (𝐶𝑅𝑆 ∨ 𝐶 = 𝑆))
25	24	notbii 319	. . . 4 ⊢ (¬ (𝑆 = 𝐶 ∨ 𝐶𝑅𝑆) ↔ ¬ (𝐶𝑅𝑆 ∨ 𝐶 = 𝑆))
26	20, 25	bitr2di 287	. . 3 ⊢ (𝜑 → (¬ (𝐶𝑅𝑆 ∨ 𝐶 = 𝑆) ↔ 𝑆𝑅𝐶))
27	13, 26	mtbird 324	. 2 ⊢ (𝜑 → ¬ ¬ (𝐶𝑅𝑆 ∨ 𝐶 = 𝑆))
28	27	notnotrd 133	1 ⊢ (𝜑 → (𝐶𝑅𝑆 ∨ 𝐶 = 𝑆))

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