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Theorem infltoreq 8950

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 6107	. . . 4 ⊢ (𝑅 Or 𝐴 ↔ ^◡𝑅 Or 𝐴)
3	1, 2	sylib 221	. . 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 8891	. . . 4 ⊢ inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, ^◡𝑅)
9	7, 8	eqtrdi 2849	. . 3 ⊢ (𝜑 → 𝑆 = sup(𝐵, 𝐴, ^◡𝑅))
10	3, 4, 5, 6, 9	supgtoreq 8918	. 2 ⊢ (𝜑 → (𝐶^◡𝑅𝑆 ∨ 𝐶 = 𝑆))
11	6	ne0d 4251	. . . . . 6 ⊢ (𝜑 → 𝐵 ≠ ∅)
12		fiinfcl 8949	. . . . . 6 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵 ⊆ 𝐴)) → inf(𝐵, 𝐴, 𝑅) ∈ 𝐵)
13	1, 5, 11, 4, 12	syl13anc 1369	. . . . 5 ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) ∈ 𝐵)
14	7, 13	eqeltrd 2890	. . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝐵)
15		brcnvg 5714	. . . . 5 ⊢ ((𝐶 ∈ 𝐵 ∧ 𝑆 ∈ 𝐵) → (𝐶^◡𝑅𝑆 ↔ 𝑆𝑅𝐶))
16	15	bicomd 226	. . . 4 ⊢ ((𝐶 ∈ 𝐵 ∧ 𝑆 ∈ 𝐵) → (𝑆𝑅𝐶 ↔ 𝐶^◡𝑅𝑆))
17	6, 14, 16	syl2anc 587	. . 3 ⊢ (𝜑 → (𝑆𝑅𝐶 ↔ 𝐶^◡𝑅𝑆))
18	17	orbi1d 914	. 2 ⊢ (𝜑 → ((𝑆𝑅𝐶 ∨ 𝐶 = 𝑆) ↔ (𝐶^◡𝑅𝑆 ∨ 𝐶 = 𝑆)))
19	10, 18	mpbird 260	1 ⊢ (𝜑 → (𝑆𝑅𝐶 ∨ 𝐶 = 𝑆))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 844 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ⊆ wss 3881 ∅c0 4243 class class class wbr 5030 Or wor 5437 ^◡ccnv 5518 Fincfn 8492 supcsup 8888 infcinf 8889

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-om 7561 df-1o 8085 df-er 8272 df-en 8493 df-fin 8496 df-sup 8890 df-inf 8891

This theorem is referenced by: (None)

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