Theorem infmin 9459

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 785	. . 3 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝐶𝑅𝑦)) → 𝐶𝑅𝑦)
6		breq1 5114	. . . 4 ⊢ (𝑧 = 𝐶 → (𝑧𝑅𝑦 ↔ 𝐶𝑅𝑦))
7	6	rspcev 3583	. . 3 ⊢ ((𝐶 ∈ 𝐵 ∧ 𝐶𝑅𝑦) → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)
8	4, 5, 7	syl2an2r 698	. 2 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝐶𝑅𝑦)) → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)
9	1, 2, 3, 8	eqinfd 9449	1 ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) = 𝐶)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∃wrex 3091   class class class wbr 5111   Or wor 5570  infcinf 9404

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-po 5571  df-so 5572  df-cnv 5671  df-iota 6496  df-riota 7373  df-sup 9405  df-inf 9406

This theorem is used by:  infpr  9468  lbinf  12179  uzinfi  12964  lcmgcdlem  16682  ramcl2lem  17087  oms0  34728  ballotlemirc  34963  inffz  36235  supinf  43043  oninfint  43996