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Theorem infltoreq 9388

Description: The infimum of a finite set is less than or equal to all the elements of the set. (Contributed by AV, 4-Sep-2020.)

**Hypotheses**
Ref	Expression
infltoreq.1	⊢ (𝜑 → 𝑅 Or 𝐴)
infltoreq.2	⊢ (𝜑 → 𝐵 ⊆ 𝐴)
infltoreq.3	⊢ (𝜑 → 𝐵 ∈ Fin)
infltoreq.4	⊢ (𝜑 → 𝐶 ∈ 𝐵)
infltoreq.5	⊢ (𝜑 → 𝑆 = inf(𝐵, 𝐴, 𝑅))

**Assertion**
Ref	Expression
infltoreq	⊢ (𝜑 → (𝑆𝑅𝐶 ∨ 𝐶 = 𝑆))

**Proof of Theorem infltoreq**
Step	Hyp	Ref	Expression
1		infltoreq.1	. . . 4 ⊢ (𝜑 → 𝑅 Or 𝐴)
2		cnvso 6235	. . . 4 ⊢ (𝑅 Or 𝐴 ↔ ^◡𝑅 Or 𝐴)
3	1, 2	sylib 218	. . 3 ⊢ (𝜑 → ^◡𝑅 Or 𝐴)
4		infltoreq.2	. . 3 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
5		infltoreq.3	. . 3 ⊢ (𝜑 → 𝐵 ∈ Fin)
6		infltoreq.4	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝐵)
7		infltoreq.5	. . . 4 ⊢ (𝜑 → 𝑆 = inf(𝐵, 𝐴, 𝑅))
8		df-inf 9327	. . . 4 ⊢ inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, ^◡𝑅)
9	7, 8	eqtrdi 2782	. . 3 ⊢ (𝜑 → 𝑆 = sup(𝐵, 𝐴, ^◡𝑅))
10	3, 4, 5, 6, 9	supgtoreq 9355	. 2 ⊢ (𝜑 → (𝐶^◡𝑅𝑆 ∨ 𝐶 = 𝑆))
11	6	ne0d 4289	. . . . . 6 ⊢ (𝜑 → 𝐵 ≠ ∅)
12		fiinfcl 9387	. . . . . 6 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵 ⊆ 𝐴)) → inf(𝐵, 𝐴, 𝑅) ∈ 𝐵)
13	1, 5, 11, 4, 12	syl13anc 1374	. . . . 5 ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ 𝐵)
14	7, 13	eqeltrd 2831	. . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝐵)
15		brcnvg 5818	. . . . 5 ⊢ ((𝐶 ∈ 𝐵 ∧ 𝑆 ∈ 𝐵) → (𝐶^◡𝑅𝑆 ↔ 𝑆𝑅𝐶))
16	15	bicomd 223	. . . 4 ⊢ ((𝐶 ∈ 𝐵 ∧ 𝑆 ∈ 𝐵) → (𝑆𝑅𝐶 ↔ 𝐶^◡𝑅𝑆))
17	6, 14, 16	syl2anc 584	. . 3 ⊢ (𝜑 → (𝑆𝑅𝐶 ↔ 𝐶^◡𝑅𝑆))
18	17	orbi1d 916	. 2 ⊢ (𝜑 → ((𝑆𝑅𝐶 ∨ 𝐶 = 𝑆) ↔ (𝐶^◡𝑅𝑆 ∨ 𝐶 = 𝑆)))
19	10, 18	mpbird 257	1 ⊢ (𝜑 → (𝑆𝑅𝐶 ∨ 𝐶 = 𝑆))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 ⊆ wss 3897 ∅c0 4280 class class class wbr 5089 Or wor 5521 ^◡ccnv 5613 Fincfn 8869 supcsup 9324 infcinf 9325

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 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-sep 5232 ax-nul 5242 ax-pr 5368 ax-un 7668

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 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4857 df-br 5090 df-opab 5152 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-om 7797 df-en 8870 df-fin 8873 df-sup 9326 df-inf 9327

This theorem is referenced by: (None)

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