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Theorem int-ineqtransd 40819
Description: InequalityTransitivity generator rule. (Contributed by Stanislas Polu, 7-Apr-2020.)
Hypotheses
Ref Expression
int-ineqtransd.1 (𝜑𝐴 ∈ ℝ)
int-ineqtransd.2 (𝜑𝐵 ∈ ℝ)
int-ineqtransd.3 (𝜑𝐶 ∈ ℝ)
int-ineqtransd.4 (𝜑𝐵𝐴)
int-ineqtransd.5 (𝜑𝐶𝐵)
Assertion
Ref Expression
int-ineqtransd (𝜑𝐶𝐴)

Proof of Theorem int-ineqtransd
StepHypRef Expression
1 int-ineqtransd.3 . 2 (𝜑𝐶 ∈ ℝ)
2 int-ineqtransd.2 . 2 (𝜑𝐵 ∈ ℝ)
3 int-ineqtransd.1 . 2 (𝜑𝐴 ∈ ℝ)
4 int-ineqtransd.5 . 2 (𝜑𝐶𝐵)
5 int-ineqtransd.4 . 2 (𝜑𝐵𝐴)
61, 2, 3, 4, 5letrd 10795 1 (𝜑𝐶𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2115   class class class wbr 5052  cr 10534  cle 10674
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-resscn 10592  ax-pre-lttri 10609  ax-pre-lttrn 10610
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-er 8285  df-en 8506  df-dom 8507  df-sdom 8508  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679
This theorem is referenced by: (None)
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