Theorem infmin 9508

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 5122	. . . 4 ⊢ (𝑧 = 𝐶 → (𝑧𝑅𝑦 ↔ 𝐶𝑅𝑦))
7	6	rspcev 3601	. . 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 9498	1 ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) = 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∃wrex 3060   class class class wbr 5119   Or wor 5560  infcinf 9453

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 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-po 5561  df-so 5562  df-cnv 5662  df-iota 6484  df-riota 7362  df-sup 9454  df-inf 9455

This theorem is referenced by:  infpr  9517  lbinf  12195  uzinfi  12944  lcmgcdlem  16625  ramcl2lem  17029  oms0  34329  ballotlemirc  34564  inffz  35747  supinf  42293  oninfint  43260