Theorem luble 18524

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 5106	. 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 18523	. . 3 ⊢ (𝜑 → (∀𝑦 ∈ 𝑆 𝑦 ≤ (𝑈‘𝑆) ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → (𝑈‘𝑆) ≤ 𝑧)))
8	7	simpld 500	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝑆 𝑦 ≤ (𝑈‘𝑆))
9		luble.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝑆)
10	1, 8, 9	rspcdva 3578	1 ⊢ (𝜑 → 𝑋 ≤ (𝑈‘𝑆))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ∀wral 3077   class class class wbr 5103  dom cdm 5651  ‘cfv 6537  Basecbs 17380  lecple 17428  lubclub 18476

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-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  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-nfc 2910  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-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7375  df-lub 18511

This theorem is used by:  ple1  18595  lubsscl  50037