Theorem inflb 8937

Description: An infimum is a lower bound. See also infcl 8936 and infglb 8938. (Contributed by AV, 3-Sep-2020.)

Hypotheses

Ref	Expression
infcl.1	⊢ (𝜑 → 𝑅 Or 𝐴)
infcl.2	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))

Assertion

Ref	Expression
inflb	⊢ (𝜑 → (𝐶 ∈ 𝐵 → ¬ 𝐶𝑅inf(𝐵, 𝐴, 𝑅)))

Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝐶(𝑥,𝑦,𝑧)

Proof of Theorem inflb

Step	Hyp	Ref	Expression
1		infcl.1	. . . . . 6 ⊢ (𝜑 → 𝑅 Or 𝐴)
2		cnvso 6107	. . . . . 6 ⊢ (𝑅 Or 𝐴 ↔ ◡𝑅 Or 𝐴)
3	1, 2	sylib 221	. . . . 5 ⊢ (𝜑 → ◡𝑅 Or 𝐴)
4		infcl.2	. . . . . 6 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))
5	1, 4	infcllem 8935	. . . . 5 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧)))
6	3, 5	supub 8907	. . . 4 ⊢ (𝜑 → (𝐶 ∈ 𝐵 → ¬ sup(𝐵, 𝐴, ◡𝑅)◡𝑅𝐶))
7	6	imp 410	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐵) → ¬ sup(𝐵, 𝐴, ◡𝑅)◡𝑅𝐶)
8		df-inf 8891	. . . . . 6 ⊢ inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, ◡𝑅)
9	8	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐵) → inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, ◡𝑅))
10	9	breq2d 5042	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐵) → (𝐶𝑅inf(𝐵, 𝐴, 𝑅) ↔ 𝐶𝑅sup(𝐵, 𝐴, ◡𝑅)))
11	3, 5	supcl 8906	. . . . 5 ⊢ (𝜑 → sup(𝐵, 𝐴, ◡𝑅) ∈ 𝐴)
12		brcnvg 5714	. . . . . 6 ⊢ ((sup(𝐵, 𝐴, ◡𝑅) ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → (sup(𝐵, 𝐴, ◡𝑅)◡𝑅𝐶 ↔ 𝐶𝑅sup(𝐵, 𝐴, ◡𝑅)))
13	12	bicomd 226	. . . . 5 ⊢ ((sup(𝐵, 𝐴, ◡𝑅) ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → (𝐶𝑅sup(𝐵, 𝐴, ◡𝑅) ↔ sup(𝐵, 𝐴, ◡𝑅)◡𝑅𝐶))
14	11, 13	sylan 583	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐵) → (𝐶𝑅sup(𝐵, 𝐴, ◡𝑅) ↔ sup(𝐵, 𝐴, ◡𝑅)◡𝑅𝐶))
15	10, 14	bitrd 282	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐵) → (𝐶𝑅inf(𝐵, 𝐴, 𝑅) ↔ sup(𝐵, 𝐴, ◡𝑅)◡𝑅𝐶))
16	7, 15	mtbird 328	. 2 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐵) → ¬ 𝐶𝑅inf(𝐵, 𝐴, 𝑅))
17	16	ex 416	1 ⊢ (𝜑 → (𝐶 ∈ 𝐵 → ¬ 𝐶𝑅inf(𝐵, 𝐴, 𝑅)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107   class class class wbr 5030   Or wor 5437  ^◡ccnv 5518  supcsup 8888  infcinf 8889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-po 5438  df-so 5439  df-cnv 5527  df-iota 6283  df-riota 7093  df-sup 8890  df-inf 8891

This theorem is referenced by:  infrelb  11613  infxrlb  12715  infssd  30472  infxrge0lb  30514  omssubadd  31668  ballotlemimin  31873  ballotlemfrcn0  31897  wsuclb  33228