Theorem inflb 9482

Description: An infimum is a lower bound. See also infcl 9481 and infglb 9483. (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 6291	. . . . . 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 9480	. . . . 5 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧)))
6	3, 5	supub 9451	. . . 4 ⊢ (𝜑 → (𝐶 ∈ 𝐵 → ¬ sup(𝐵, 𝐴, ◡𝑅)◡𝑅𝐶))
7	6	imp 412	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐵) → ¬ sup(𝐵, 𝐴, ◡𝑅)◡𝑅𝐶)
8		df-inf 9435	. . . . . 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 5115	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐵) → (𝐶𝑅inf(𝐵, 𝐴, 𝑅) ↔ 𝐶𝑅sup(𝐵, 𝐴, ◡𝑅)))
11	3, 5	supcl 9450	. . . . 5 ⊢ (𝜑 → sup(𝐵, 𝐴, ◡𝑅) ∈ 𝐴)
12		brcnvg 5857	. . . . . 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 592	. . . 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 418	1 ⊢ (𝜑 → (𝐶 ∈ 𝐵 → ¬ 𝐶𝑅inf(𝐵, 𝐴, 𝑅)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087   class class class wbr 5103   Or wor 5558  ^◡ccnv 5650  supcsup 9432  infcinf 9433

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-pr 5391

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-po 5559  df-so 5560  df-cnv 5659  df-iota 6494  df-riota 7377  df-sup 9434  df-inf 9435

This theorem is used by:  infssd  9486  infrelb  12302  infxrlb  13465  infxrge0lb  33356  omssubadd  34932  ballotlemimin  35138  ballotlemfrcn0  35162  wsuclb  36590