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Theorem pleval2i 17279
Description: One direction of pleval2 17280. (Contributed by Mario Carneiro, 8-Feb-2015.)
Hypotheses
Ref Expression
pleval2.b 𝐵 = (Base‘𝐾)
pleval2.l = (le‘𝐾)
pleval2.s < = (lt‘𝐾)
Assertion
Ref Expression
pleval2i ((𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → (𝑋 < 𝑌𝑋 = 𝑌)))

Proof of Theorem pleval2i
StepHypRef Expression
1 elfvdm 6443 . . . . . . . . 9 (𝑋 ∈ (Base‘𝐾) → 𝐾 ∈ dom Base)
2 pleval2.b . . . . . . . . 9 𝐵 = (Base‘𝐾)
31, 2eleq2s 2896 . . . . . . . 8 (𝑋𝐵𝐾 ∈ dom Base)
43adantr 473 . . . . . . 7 ((𝑋𝐵𝑌𝐵) → 𝐾 ∈ dom Base)
5 pleval2.l . . . . . . . . 9 = (le‘𝐾)
6 pleval2.s . . . . . . . . 9 < = (lt‘𝐾)
75, 6pltval 17275 . . . . . . . 8 ((𝐾 ∈ dom Base ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ (𝑋 𝑌𝑋𝑌)))
873expb 1150 . . . . . . 7 ((𝐾 ∈ dom Base ∧ (𝑋𝐵𝑌𝐵)) → (𝑋 < 𝑌 ↔ (𝑋 𝑌𝑋𝑌)))
94, 8mpancom 680 . . . . . 6 ((𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ (𝑋 𝑌𝑋𝑌)))
109biimpar 470 . . . . 5 (((𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌𝑋𝑌)) → 𝑋 < 𝑌)
1110expr 449 . . . 4 (((𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) → (𝑋𝑌𝑋 < 𝑌))
1211necon1bd 2989 . . 3 (((𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) → (¬ 𝑋 < 𝑌𝑋 = 𝑌))
1312orrd 890 . 2 (((𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) → (𝑋 < 𝑌𝑋 = 𝑌))
1413ex 402 1 ((𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → (𝑋 < 𝑌𝑋 = 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  wo 874   = wceq 1653  wcel 2157  wne 2971   class class class wbr 4843  dom cdm 5312  cfv 6101  Basecbs 16184  lecple 16274  ltcplt 17256
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-iota 6064  df-fun 6103  df-fv 6109  df-plt 17273
This theorem is referenced by:  pleval2  17280  pospo  17288
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