Theorem infnlb 9402

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 9400	. . . . 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 3056	. . . 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 2109  ∀wral 3044  ∃wrex 3053   class class class wbr 5095   Or wor 5530  infcinf 9350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-po 5531  df-so 5532  df-cnv 5631  df-iota 6442  df-riota 7310  df-sup 9351  df-inf 9352

This theorem is referenced by:  infssd  9403