Theorem infmin 9380

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 5094	. . . 4 ⊢ (𝑧 = 𝐶 → (𝑧𝑅𝑦 ↔ 𝐶𝑅𝑦))
7	6	rspcev 3577	. . 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 9370	1 ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) = 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∃wrex 3056   class class class wbr 5091   Or wor 5523  infcinf 9325

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-po 5524  df-so 5525  df-cnv 5624  df-iota 6437  df-riota 7303  df-sup 9326  df-inf 9327

This theorem is referenced by:  infpr  9389  lbinf  12075  uzinfi  12826  lcmgcdlem  16517  ramcl2lem  16921  oms0  34308  ballotlemirc  34543  inffz  35772  supinf  42281  oninfint  43275