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Theorem pltfval 16782
 Description: Value of the less-than relation. (Contributed by Mario Carneiro, 8-Feb-2015.)
Hypotheses
Ref Expression
pltval.l = (le‘𝐾)
pltval.s < = (lt‘𝐾)
Assertion
Ref Expression
pltfval (𝐾𝐴< = ( ∖ I ))

Proof of Theorem pltfval
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 pltval.s . 2 < = (lt‘𝐾)
2 elex 3185 . . 3 (𝐾𝐴𝐾 ∈ V)
3 fveq2 6103 . . . . . 6 (𝑝 = 𝐾 → (le‘𝑝) = (le‘𝐾))
4 pltval.l . . . . . 6 = (le‘𝐾)
53, 4syl6eqr 2662 . . . . 5 (𝑝 = 𝐾 → (le‘𝑝) = )
65difeq1d 3689 . . . 4 (𝑝 = 𝐾 → ((le‘𝑝) ∖ I ) = ( ∖ I ))
7 df-plt 16781 . . . 4 lt = (𝑝 ∈ V ↦ ((le‘𝑝) ∖ I ))
8 fvex 6113 . . . . . 6 (le‘𝐾) ∈ V
94, 8eqeltri 2684 . . . . 5 ∈ V
10 difexg 4735 . . . . 5 ( ∈ V → ( ∖ I ) ∈ V)
119, 10ax-mp 5 . . . 4 ( ∖ I ) ∈ V
126, 7, 11fvmpt 6191 . . 3 (𝐾 ∈ V → (lt‘𝐾) = ( ∖ I ))
132, 12syl 17 . 2 (𝐾𝐴 → (lt‘𝐾) = ( ∖ I ))
141, 13syl5eq 2656 1 (𝐾𝐴< = ( ∖ I ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∖ cdif 3537   I cid 4948  ‘cfv 5804  lecple 15775  ltcplt 16764 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-plt 16781 This theorem is referenced by:  pltval  16783  opsrtoslem2  19306  relt  19780  oppglt  28985  xrslt  29007  submarchi  29071
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