Theorem infnlb 9400

Description: A lower bound is not greater than the infimum. (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
infnlb	⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐵 ¬ 𝑧𝑅𝐶) → ¬ inf(𝐵, 𝐴, 𝑅)𝑅𝐶))

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

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

Proof of Theorem infnlb

Step	Hyp	Ref	Expression
1		infcl.1	. . . . . 6 ⊢ (𝜑 → 𝑅 Or 𝐴)
2		infcl.2	. . . . . 6 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))
3	1, 2	infglb 9398	. . . . 5 ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ inf(𝐵, 𝐴, 𝑅)𝑅𝐶) → ∃𝑧 ∈ 𝐵 𝑧𝑅𝐶))
4	3	expdimp 452	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (inf(𝐵, 𝐴, 𝑅)𝑅𝐶 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝐶))
5		dfrex2 3064	. . . 4 ⊢ (∃𝑧 ∈ 𝐵 𝑧𝑅𝐶 ↔ ¬ ∀𝑧 ∈ 𝐵 ¬ 𝑧𝑅𝐶)
6	4, 5	imbitrdi 251	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (inf(𝐵, 𝐴, 𝑅)𝑅𝐶 → ¬ ∀𝑧 ∈ 𝐵 ¬ 𝑧𝑅𝐶))
7	6	con2d 134	. 2 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (∀𝑧 ∈ 𝐵 ¬ 𝑧𝑅𝐶 → ¬ inf(𝐵, 𝐴, 𝑅)𝑅𝐶))
8	7	expimpd 453	1 ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ ∀𝑧 ∈ 𝐵 ¬ 𝑧𝑅𝐶) → ¬ inf(𝐵, 𝐴, 𝑅)𝑅𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∈ wcel 2114  ∀wral 3052  ∃wrex 3061   class class class wbr 5099   Or wor 5532  infcinf 9348

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-po 5533  df-so 5534  df-cnv 5633  df-iota 6449  df-riota 7317  df-sup 9349  df-inf 9350

This theorem is referenced by:  infssd  9401