Theorem luble 18311

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 5151	. 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 18310	. . 3 ⊢ (𝜑 → (∀𝑦 ∈ 𝑆 𝑦 ≤ (𝑈‘𝑆) ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → (𝑈‘𝑆) ≤ 𝑧)))
8	7	simpld 495	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝑆 𝑦 ≤ (𝑈‘𝑆))
9		luble.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝑆)
10	1, 8, 9	rspcdva 3613	1 ⊢ (𝜑 → 𝑋 ≤ (𝑈‘𝑆))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2106  ∀wral 3061   class class class wbr 5148  dom cdm 5676  ‘cfv 6543  Basecbs 17143  lecple 17203  lubclub 18261

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7364  df-lub 18298

This theorem is referenced by:  ple1  18382  lubsscl  47583