Theorem infmin 9534

Description: The smallest element of a set is its infimum. Note that the converse is not true; the infimum might not be an element of the set considered. (Contributed by AV, 3-Sep-2020.)

Hypotheses

Ref	Expression
infmin.1	⊢ (𝜑 → 𝑅 Or 𝐴)
infmin.2	⊢ (𝜑 → 𝐶 ∈ 𝐴)
infmin.3	⊢ (𝜑 → 𝐶 ∈ 𝐵)
infmin.4	⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ¬ 𝑦𝑅𝐶)

Assertion

Ref	Expression
infmin	⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) = 𝐶)

Distinct variable groups:   𝑦,𝐴   𝑦,𝐵   𝑦,𝐶   𝑦,𝑅   𝜑,𝑦

Proof of Theorem infmin

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		infmin.1	. 2 ⊢ (𝜑 → 𝑅 Or 𝐴)
2		infmin.2	. 2 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
3		infmin.4	. 2 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ¬ 𝑦𝑅𝐶)
4		infmin.3	. . 3 ⊢ (𝜑 → 𝐶 ∈ 𝐵)
5		simprr 773	. . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝐶𝑅𝑦)) → 𝐶𝑅𝑦)
6		breq1 5146	. . . 4 ⊢ (𝑧 = 𝐶 → (𝑧𝑅𝑦 ↔ 𝐶𝑅𝑦))
7	6	rspcev 3622	. . 3 ⊢ ((𝐶 ∈ 𝐵 ∧ 𝐶𝑅𝑦) → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)
8	4, 5, 7	syl2an2r 685	. 2 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝐶𝑅𝑦)) → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)
9	1, 2, 3, 8	eqinfd 9525	1 ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) = 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∃wrex 3070   class class class wbr 5143   Or wor 5591  infcinf 9481

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 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-po 5592  df-so 5593  df-cnv 5693  df-iota 6514  df-riota 7388  df-sup 9482  df-inf 9483

This theorem is referenced by:  infpr  9543  lbinf  12221  uzinfi  12970  lcmgcdlem  16643  ramcl2lem  17047  oms0  34299  ballotlemirc  34534  inffz  35730  supinf  42283  oninfint  43248