Theorem supgtoreq 9372

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 4292	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ≠ ∅)
7		fisup2g 9370	. . . . . . . 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 1374	. . . . . . 7 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))
9		ssrexv 4001	. . . . . . 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 9360	. . . . 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 5114	. . 3 ⊢ (𝜑 → ¬ 𝑆𝑅𝐶)
14		fisupcl 9371	. . . . . . . 8 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵 ⊆ 𝐴)) → sup(𝐵, 𝐴, 𝑅) ∈ 𝐵)
15	3, 5, 6, 4, 14	syl13anc 1374	. . . . . . 7 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐵)
16	4, 15	sseldd 3932	. . . . . 6 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)
17	1, 16	eqeltrd 2834	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ 𝐴)
18	4, 2	sseldd 3932	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
19		sotric 5560	. . . . 5 ⊢ ((𝑅 Or 𝐴 ∧ (𝑆 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝑆𝑅𝐶 ↔ ¬ (𝑆 = 𝐶 ∨ 𝐶𝑅𝑆)))
20	3, 17, 18, 19	syl12anc 836	. . . 4 ⊢ (𝜑 → (𝑆𝑅𝐶 ↔ ¬ (𝑆 = 𝐶 ∨ 𝐶𝑅𝑆)))
21		orcom 870	. . . . . 6 ⊢ ((𝑆 = 𝐶 ∨ 𝐶𝑅𝑆) ↔ (𝐶𝑅𝑆 ∨ 𝑆 = 𝐶))
22		eqcom 2741	. . . . . . 7 ⊢ (𝑆 = 𝐶 ↔ 𝐶 = 𝑆)
23	22	orbi2i 912	. . . . . 6 ⊢ ((𝐶𝑅𝑆 ∨ 𝑆 = 𝐶) ↔ (𝐶𝑅𝑆 ∨ 𝐶 = 𝑆))
24	21, 23	bitri 275	. . . . 5 ⊢ ((𝑆 = 𝐶 ∨ 𝐶𝑅𝑆) ↔ (𝐶𝑅𝑆 ∨ 𝐶 = 𝑆))
25	24	notbii 320	. . . 4 ⊢ (¬ (𝑆 = 𝐶 ∨ 𝐶𝑅𝑆) ↔ ¬ (𝐶𝑅𝑆 ∨ 𝐶 = 𝑆))
26	20, 25	bitr2di 288	. . 3 ⊢ (𝜑 → (¬ (𝐶𝑅𝑆 ∨ 𝐶 = 𝑆) ↔ 𝑆𝑅𝐶))
27	13, 26	mtbird 325	. 2 ⊢ (𝜑 → ¬ ¬ (𝐶𝑅𝑆 ∨ 𝐶 = 𝑆))
28	27	notnotrd 133	1 ⊢ (𝜑 → (𝐶𝑅𝑆 ∨ 𝐶 = 𝑆))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2113   ≠ wne 2930  ∀wral 3049  ∃wrex 3058   ⊆ wss 3899  ∅c0 4283   class class class wbr 5096   Or wor 5529  Fincfn 8881  supcsup 9341

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-riota 7313  df-om 7807  df-en 8882  df-fin 8885  df-sup 9343

This theorem is referenced by:  infltoreq  9405  supfirege  12127  safesnsupfilb  43601