Theorem infempty 9521

Description: The infimum of an empty set under a base set which has a unique greatest element is the greatest element of the base set. (Contributed by AV, 4-Sep-2020.)

Assertion

Ref	Expression
infempty	⊢ ((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑋𝑅𝑦) ∧ ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦) → inf(∅, 𝐴, 𝑅) = 𝑋)

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝑅,𝑦   𝑥,𝑋,𝑦

Proof of Theorem infempty

Step	Hyp	Ref	Expression
1		df-inf 9455	. 2 ⊢ inf(∅, 𝐴, 𝑅) = sup(∅, 𝐴, ◡𝑅)
2		cnvso 6277	. . 3 ⊢ (𝑅 Or 𝐴 ↔ ◡𝑅 Or 𝐴)
3		brcnvg 5859	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑦◡𝑅𝑋 ↔ 𝑋𝑅𝑦))
4	3	ancoms 458	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑦◡𝑅𝑋 ↔ 𝑋𝑅𝑦))
5	4	bicomd 223	. . . . . 6 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑋𝑅𝑦 ↔ 𝑦◡𝑅𝑋))
6	5	notbid 318	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (¬ 𝑋𝑅𝑦 ↔ ¬ 𝑦◡𝑅𝑋))
7	6	ralbidva 3161	. . . 4 ⊢ (𝑋 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 ¬ 𝑋𝑅𝑦 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑦◡𝑅𝑋))
8	7	pm5.32i 574	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑋𝑅𝑦) ↔ (𝑋 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑦◡𝑅𝑋))
9		brcnvg 5859	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦))
10	9	ancoms 458	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦))
11	10	bicomd 223	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ↔ 𝑦◡𝑅𝑥))
12	11	notbid 318	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑦◡𝑅𝑥))
13	12	ralbidva 3161	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑦◡𝑅𝑥))
14	13	reubiia 3366	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦 ↔ ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦◡𝑅𝑥)
15		sup0 9479	. . 3 ⊢ ((◡𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑦◡𝑅𝑋) ∧ ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦◡𝑅𝑥) → sup(∅, 𝐴, ◡𝑅) = 𝑋)
16	2, 8, 14, 15	syl3anb 1161	. 2 ⊢ ((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑋𝑅𝑦) ∧ ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦) → sup(∅, 𝐴, ◡𝑅) = 𝑋)
17	1, 16	eqtrid 2782	1 ⊢ ((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑋𝑅𝑦) ∧ ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦) → inf(∅, 𝐴, 𝑅) = 𝑋)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108  ∀wral 3051  ∃!wreu 3357  ∅c0 4308   class class class wbr 5119   Or wor 5560  ^◡ccnv 5653  supcsup 9452  infcinf 9453

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-po 5561  df-so 5562  df-cnv 5662  df-iota 6484  df-riota 7362  df-sup 9454  df-inf 9455

This theorem is referenced by: (None)