Theorem supgtoreq 9398

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 4301	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ≠ ∅)
7		fisup2g 9396	. . . . . . . 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 4013	. . . . . . 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 9386	. . . . 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 5120	. . 3 ⊢ (𝜑 → ¬ 𝑆𝑅𝐶)
14		fisupcl 9397	. . . . . . . 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 3944	. . . . . 6 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)
17	1, 16	eqeltrd 2828	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ 𝐴)
18	4, 2	sseldd 3944	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
19		sotric 5569	. . . . 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 2736	. . . . . . 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 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  ∃wrex 3053   ⊆ wss 3911  ∅c0 4292   class class class wbr 5102   Or wor 5538  Fincfn 8895  supcsup 9367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-tr 5210  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-ord 6323  df-on 6324  df-lim 6325  df-suc 6326  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-riota 7326  df-om 7823  df-en 8896  df-fin 8899  df-sup 9369

This theorem is referenced by:  infltoreq  9431  supfirege  12146  safesnsupfilb  43380