Theorem supgtoreq 9386

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 4296	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ≠ ∅)
7		fisup2g 9384	. . . . . . . 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 1375	. . . . . . 7 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))
9		ssrexv 4005	. . . . . . 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 9374	. . . . 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 5118	. . 3 ⊢ (𝜑 → ¬ 𝑆𝑅𝐶)
14		fisupcl 9385	. . . . . . . 8 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵 ⊆ 𝐴)) → sup(𝐵, 𝐴, 𝑅) ∈ 𝐵)
15	3, 5, 6, 4, 14	syl13anc 1375	. . . . . . 7 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐵)
16	4, 15	sseldd 3936	. . . . . 6 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)
17	1, 16	eqeltrd 2837	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ 𝐴)
18	4, 2	sseldd 3936	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
19		sotric 5570	. . . . 5 ⊢ ((𝑅 Or 𝐴 ∧ (𝑆 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝑆𝑅𝐶 ↔ ¬ (𝑆 = 𝐶 ∨ 𝐶𝑅𝑆)))
20	3, 17, 18, 19	syl12anc 837	. . . 4 ⊢ (𝜑 → (𝑆𝑅𝐶 ↔ ¬ (𝑆 = 𝐶 ∨ 𝐶𝑅𝑆)))
21		orcom 871	. . . . . 6 ⊢ ((𝑆 = 𝐶 ∨ 𝐶𝑅𝑆) ↔ (𝐶𝑅𝑆 ∨ 𝑆 = 𝐶))
22		eqcom 2744	. . . . . . 7 ⊢ (𝑆 = 𝐶 ↔ 𝐶 = 𝑆)
23	22	orbi2i 913	. . . . . 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 848   = wceq 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  ∃wrex 3062   ⊆ wss 3903  ∅c0 4287   class class class wbr 5100   Or wor 5539  Fincfn 8895  supcsup 9355

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rmo 3352  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-om 7819  df-en 8896  df-fin 8899  df-sup 9357

This theorem is referenced by:  infltoreq  9419  supfirege  12141  safesnsupfilb  43774