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Theorem pleval2i 17011
Description: One direction of pleval2 17012. (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 6258 . . . . . . . . 9 (𝑋 ∈ (Base‘𝐾) → 𝐾 ∈ dom Base)
2 pleval2.b . . . . . . . . 9 𝐵 = (Base‘𝐾)
31, 2eleq2s 2748 . . . . . . . 8 (𝑋𝐵𝐾 ∈ dom Base)
43adantr 480 . . . . . . 7 ((𝑋𝐵𝑌𝐵) → 𝐾 ∈ dom Base)
5 pleval2.l . . . . . . . . 9 = (le‘𝐾)
6 pleval2.s . . . . . . . . 9 < = (lt‘𝐾)
75, 6pltval 17007 . . . . . . . 8 ((𝐾 ∈ dom Base ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ (𝑋 𝑌𝑋𝑌)))
873expb 1285 . . . . . . 7 ((𝐾 ∈ dom Base ∧ (𝑋𝐵𝑌𝐵)) → (𝑋 < 𝑌 ↔ (𝑋 𝑌𝑋𝑌)))
94, 8mpancom 704 . . . . . 6 ((𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ (𝑋 𝑌𝑋𝑌)))
109biimpar 501 . . . . 5 (((𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌𝑋𝑌)) → 𝑋 < 𝑌)
1110expr 642 . . . 4 (((𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) → (𝑋𝑌𝑋 < 𝑌))
1211necon1bd 2841 . . 3 (((𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) → (¬ 𝑋 < 𝑌𝑋 = 𝑌))
1312orrd 392 . 2 (((𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) → (𝑋 < 𝑌𝑋 = 𝑌))
1413ex 449 1 ((𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → (𝑋 < 𝑌𝑋 = 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383   = wceq 1523  wcel 2030  wne 2823   class class class wbr 4685  dom cdm 5143  cfv 5926  Basecbs 15904  lecple 15995  ltcplt 16988
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-plt 17005
This theorem is referenced by:  pleval2  17012  pospo  17020
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