Theorem supgtoreq 9506

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 4335	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ≠ ∅)
7		fisup2g 9504	. . . . . . . 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 1369	. . . . . . 7 ⊢ (𝜑 → ∃𝑥 ∈ 𝐵 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))
9		ssrexv 4048	. . . . . . 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 9495	. . . . 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 5163	. . 3 ⊢ (𝜑 → ¬ 𝑆𝑅𝐶)
14		fisupcl 9505	. . . . . . . 8 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵 ⊆ 𝐴)) → sup(𝐵, 𝐴, 𝑅) ∈ 𝐵)
15	3, 5, 6, 4, 14	syl13anc 1369	. . . . . . 7 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐵)
16	4, 15	sseldd 3979	. . . . . 6 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ 𝐴)
17	1, 16	eqeltrd 2826	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ 𝐴)
18	4, 2	sseldd 3979	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
19		sotric 5614	. . . . 5 ⊢ ((𝑅 Or 𝐴 ∧ (𝑆 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → (𝑆𝑅𝐶 ↔ ¬ (𝑆 = 𝐶 ∨ 𝐶𝑅𝑆)))
20	3, 17, 18, 19	syl12anc 835	. . . 4 ⊢ (𝜑 → (𝑆𝑅𝐶 ↔ ¬ (𝑆 = 𝐶 ∨ 𝐶𝑅𝑆)))
21		orcom 868	. . . . . 6 ⊢ ((𝑆 = 𝐶 ∨ 𝐶𝑅𝑆) ↔ (𝐶𝑅𝑆 ∨ 𝑆 = 𝐶))
22		eqcom 2733	. . . . . . 7 ⊢ (𝑆 = 𝐶 ↔ 𝐶 = 𝑆)
23	22	orbi2i 910	. . . . . 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 394   ∨ wo 845   = wceq 1534   ∈ wcel 2099   ≠ wne 2930  ∀wral 3051  ∃wrex 3060   ⊆ wss 3946  ∅c0 4322   class class class wbr 5145   Or wor 5585  Fincfn 8966  supcsup 9476

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pr 5425  ax-un 7738

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-br 5146  df-opab 5208  df-tr 5263  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-riota 7372  df-om 7869  df-en 8967  df-fin 8970  df-sup 9478

This theorem is referenced by:  infltoreq  9538  supfirege  12247  safesnsupfilb  43122