Theorem infmin 9454

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 772	. . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝐶𝑅𝑦)) → 𝐶𝑅𝑦)
6		breq1 5113	. . . 4 ⊢ (𝑧 = 𝐶 → (𝑧𝑅𝑦 ↔ 𝐶𝑅𝑦))
7	6	rspcev 3591	. . 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 9444	1 ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) = 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3054   class class class wbr 5110   Or wor 5548  infcinf 9399

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

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 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-po 5549  df-so 5550  df-cnv 5649  df-iota 6467  df-riota 7347  df-sup 9400  df-inf 9401

This theorem is referenced by:  infpr  9463  lbinf  12143  uzinfi  12894  lcmgcdlem  16583  ramcl2lem  16987  oms0  34295  ballotlemirc  34530  inffz  35724  supinf  42237  oninfint  43232