Theorem infempty 9470

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 9404	. 2 ⊢ inf(∅, 𝐴, 𝑅) = sup(∅, 𝐴, ◡𝑅)
2		cnvso 6291	. . 3 ⊢ (𝑅 Or 𝐴 ↔ ◡𝑅 Or 𝐴)
3		brcnvg 5867	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑦◡𝑅𝑋 ↔ 𝑋𝑅𝑦))
4	3	ancoms 463	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑦◡𝑅𝑋 ↔ 𝑋𝑅𝑦))
5	4	bicomd 226	. . . . . 6 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑋𝑅𝑦 ↔ 𝑦◡𝑅𝑋))
6	5	notbid 321	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (¬ 𝑋𝑅𝑦 ↔ ¬ 𝑦◡𝑅𝑋))
7	6	ralbidva 3186	. . . 4 ⊢ (𝑋 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 ¬ 𝑋𝑅𝑦 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑦◡𝑅𝑋))
8	7	pm5.32i 584	. . 3 ⊢ ((𝑋 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑋𝑅𝑦) ↔ (𝑋 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑦◡𝑅𝑋))
9		brcnvg 5867	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦))
10	9	ancoms 463	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦))
11	10	bicomd 226	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ↔ 𝑦◡𝑅𝑥))
12	11	notbid 321	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑦◡𝑅𝑥))
13	12	ralbidva 3186	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦 ↔ ∀𝑦 ∈ 𝐴 ¬ 𝑦◡𝑅𝑥))
14	13	reubiia 3376	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦 ↔ ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑦◡𝑅𝑥)
15		sup0 9428	. . 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 1179	. 2 ⊢ ((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ¬ 𝑋𝑅𝑦) ∧ ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥𝑅𝑦) → sup(∅, 𝐴, ◡𝑅) = 𝑋)
17	1, 16	eqtrid 2810	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 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ∃!wreu 3367  ∅c0 4287   class class class wbr 5110   Or wor 5570  ^◡ccnv 5662  supcsup 9401  infcinf 9402

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-po 5571  df-so 5572  df-cnv 5671  df-iota 6494  df-riota 7369  df-sup 9403  df-inf 9404

This theorem is referenced by: (None)