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Theorem p0le 17165
Description: Any element is less than or equal to a poset's upper bound (if defined). (Contributed by NM, 22-Oct-2011.) (Revised by NM, 13-Sep-2018.)
Hypotheses
Ref Expression
p0le.b 𝐵 = (Base‘𝐾)
p0le.g 𝐺 = (glb‘𝐾)
p0le.l = (le‘𝐾)
p0le.0 0 = (0.‘𝐾)
p0le.k (𝜑𝐾𝑉)
p0le.x (𝜑𝑋𝐵)
p0le.d (𝜑𝐵 ∈ dom 𝐺)
Assertion
Ref Expression
p0le (𝜑0 𝑋)

Proof of Theorem p0le
StepHypRef Expression
1 p0le.k . . 3 (𝜑𝐾𝑉)
2 p0le.b . . . 4 𝐵 = (Base‘𝐾)
3 p0le.g . . . 4 𝐺 = (glb‘𝐾)
4 p0le.0 . . . 4 0 = (0.‘𝐾)
52, 3, 4p0val 17163 . . 3 (𝐾𝑉0 = (𝐺𝐵))
61, 5syl 17 . 2 (𝜑0 = (𝐺𝐵))
7 p0le.l . . 3 = (le‘𝐾)
8 p0le.d . . 3 (𝜑𝐵 ∈ dom 𝐺)
9 p0le.x . . 3 (𝜑𝑋𝐵)
102, 7, 3, 1, 8, 9glble 17122 . 2 (𝜑 → (𝐺𝐵) 𝑋)
116, 10eqbrtrd 4782 1 (𝜑0 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1596  wcel 2103   class class class wbr 4760  dom cdm 5218  cfv 6001  Basecbs 15980  lecple 16071  glbcglb 17065  0.cp0 17159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-8 2105  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704  ax-rep 4879  ax-sep 4889  ax-nul 4897  ax-pow 4948  ax-pr 5011
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-eu 2575  df-mo 2576  df-clab 2711  df-cleq 2717  df-clel 2720  df-nfc 2855  df-ne 2897  df-ral 3019  df-rex 3020  df-reu 3021  df-rab 3023  df-v 3306  df-sbc 3542  df-csb 3640  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-pw 4268  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  df-iun 4630  df-br 4761  df-opab 4821  df-mpt 4838  df-id 5128  df-xp 5224  df-rel 5225  df-cnv 5226  df-co 5227  df-dm 5228  df-rn 5229  df-res 5230  df-ima 5231  df-iota 5964  df-fun 6003  df-fn 6004  df-f 6005  df-f1 6006  df-fo 6007  df-f1o 6008  df-fv 6009  df-riota 6726  df-glb 17097  df-p0 17161
This theorem is referenced by:  op0le  34893  atl0le  35011
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