Theorem inflb 8953

Description: An infimum is a lower bound. See also infcl 8952 and infglb 8954. (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 6139	. . . . . 6 ⊢ (𝑅 Or 𝐴 ↔ ◡𝑅 Or 𝐴)
3	1, 2	sylib 220	. . . . 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 8951	. . . . 5 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧)))
6	3, 5	supub 8923	. . . 4 ⊢ (𝜑 → (𝐶 ∈ 𝐵 → ¬ sup(𝐵, 𝐴, ◡𝑅)◡𝑅𝐶))
7	6	imp 409	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐵) → ¬ sup(𝐵, 𝐴, ◡𝑅)◡𝑅𝐶)
8		df-inf 8907	. . . . . 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 5078	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐵) → (𝐶𝑅inf(𝐵, 𝐴, 𝑅) ↔ 𝐶𝑅sup(𝐵, 𝐴, ◡𝑅)))
11	3, 5	supcl 8922	. . . . 5 ⊢ (𝜑 → sup(𝐵, 𝐴, ◡𝑅) ∈ 𝐴)
12		brcnvg 5750	. . . . . 6 ⊢ ((sup(𝐵, 𝐴, ◡𝑅) ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → (sup(𝐵, 𝐴, ◡𝑅)◡𝑅𝐶 ↔ 𝐶𝑅sup(𝐵, 𝐴, ◡𝑅)))
13	12	bicomd 225	. . . . 5 ⊢ ((sup(𝐵, 𝐴, ◡𝑅) ∈ 𝐴 ∧ 𝐶 ∈ 𝐵) → (𝐶𝑅sup(𝐵, 𝐴, ◡𝑅) ↔ sup(𝐵, 𝐴, ◡𝑅)◡𝑅𝐶))
14	11, 13	sylan 582	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐵) → (𝐶𝑅sup(𝐵, 𝐴, ◡𝑅) ↔ sup(𝐵, 𝐴, ◡𝑅)◡𝑅𝐶))
15	10, 14	bitrd 281	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐵) → (𝐶𝑅inf(𝐵, 𝐴, 𝑅) ↔ sup(𝐵, 𝐴, ◡𝑅)◡𝑅𝐶))
16	7, 15	mtbird 327	. 2 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐵) → ¬ 𝐶𝑅inf(𝐵, 𝐴, 𝑅))
17	16	ex 415	1 ⊢ (𝜑 → (𝐶 ∈ 𝐵 → ¬ 𝐶𝑅inf(𝐵, 𝐴, 𝑅)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3138  ∃wrex 3139   class class class wbr 5066   Or wor 5473  ^◡ccnv 5554  supcsup 8904  infcinf 8905

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pr 5330

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-po 5474  df-so 5475  df-cnv 5563  df-iota 6314  df-riota 7114  df-sup 8906  df-inf 8907

This theorem is referenced by:  infrelb  11626  infxrlb  12728  infssd  30446  infxrge0lb  30488  omssubadd  31558  ballotlemimin  31763  ballotlemfrcn0  31787  wsuclb  33115