Theorem infempty 8824

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 8760	. 2 ⊢ inf(∅, 𝐴, 𝑅) = sup(∅, 𝐴, ◡𝑅)
2		cnvso 6021	. . 3 ⊢ (𝑅 Or 𝐴 ↔ ◡𝑅 Or 𝐴)
3		brcnvg 5643	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑦◡𝑅𝑋 ↔ 𝑋𝑅𝑦))
4	3	ancoms 459	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑦◡𝑅𝑋 ↔ 𝑋𝑅𝑦))
5	4	bicomd 224	. . . . . 6 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑋𝑅𝑦 ↔ 𝑦◡𝑅𝑋))
6	5	notbid 319	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (¬ 𝑋𝑅𝑦 ↔ ¬ 𝑦◡𝑅𝑋))
7	6	ralbidva 3165	. . . 4 ⊢ (𝑋 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 ¬ 𝑋𝑅𝑦 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑦◡𝑅𝑋))
8	7	pm5.32i 575	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑋𝑅𝑦) ↔ (𝑋 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑦◡𝑅𝑋))
9		brcnvg 5643	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦))
10	9	ancoms 459	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦))
11	10	bicomd 224	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ↔ 𝑦◡𝑅𝑥))
12	11	notbid 319	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑦◡𝑅𝑥))
13	12	ralbidva 3165	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑦◡𝑅𝑥))
14	13	reubiia 3352	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦 ↔ ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦◡𝑅𝑥)
15		sup0 8783	. . 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 1154	. 2 ⊢ ((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑋𝑅𝑦) ∧ ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦) → sup(∅, 𝐴, ◡𝑅) = 𝑋)
17	1, 16	syl5eq 2845	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 207   ∧ wa 396   ∧ w3a 1080   = wceq 1525   ∈ wcel 2083  ∀wral 3107  ∃!wreu 3109  ∅c0 4217   class class class wbr 4968   Or wor 5368  ^◡ccnv 5449  supcsup 8757  infcinf 8758

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pr 5228

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-reu 3114  df-rmo 3115  df-rab 3116  df-v 3442  df-sbc 3712  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-br 4969  df-opab 5031  df-po 5369  df-so 5370  df-cnv 5458  df-iota 6196  df-riota 6984  df-sup 8759  df-inf 8760

This theorem is referenced by: (None)