Theorem luble 18271

Description: The greatest lower bound is the least element. (Contributed by NM, 22-Oct-2011.) (Revised by NM, 7-Sep-2018.)

Hypotheses

Ref	Expression
lubprop.b	⊢ 𝐵 = (Base‘𝐾)
lubprop.l	⊢ ≤ = (le‘𝐾)
lubprop.u	⊢ 𝑈 = (lub‘𝐾)
lubprop.k	⊢ (𝜑 → 𝐾 ∈ 𝑉)
lubprop.s	⊢ (𝜑 → 𝑆 ∈ dom 𝑈)
luble.x	⊢ (𝜑 → 𝑋 ∈ 𝑆)

Assertion

Ref	Expression
luble	⊢ (𝜑 → 𝑋 ≤ (𝑈‘𝑆))

Proof of Theorem luble

Dummy variables 𝑧 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq1 5098	. 2 ⊢ (𝑦 = 𝑋 → (𝑦 ≤ (𝑈‘𝑆) ↔ 𝑋 ≤ (𝑈‘𝑆)))
2		lubprop.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
3		lubprop.l	. . . 4 ⊢ ≤ = (le‘𝐾)
4		lubprop.u	. . . 4 ⊢ 𝑈 = (lub‘𝐾)
5		lubprop.k	. . . 4 ⊢ (𝜑 → 𝐾 ∈ 𝑉)
6		lubprop.s	. . . 4 ⊢ (𝜑 → 𝑆 ∈ dom 𝑈)
7	2, 3, 4, 5, 6	lubprop 18270	. . 3 ⊢ (𝜑 → (∀𝑦 ∈ 𝑆 𝑦 ≤ (𝑈‘𝑆) ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → (𝑈‘𝑆) ≤ 𝑧)))
8	7	simpld 494	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝑆 𝑦 ≤ (𝑈‘𝑆))
9		luble.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝑆)
10	1, 8, 9	rspcdva 3574	1 ⊢ (𝜑 → 𝑋 ≤ (𝑈‘𝑆))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  ∀wral 3048   class class class wbr 5095  dom cdm 5621  ‘cfv 6489  Basecbs 17127  lecple 17175  lubclub 18223

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7312  df-lub 18258

This theorem is referenced by:  ple1  18342  lubsscl  49121