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Theorem pleval2i 17086
Description: One direction of pleval2 17087. (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 6333 . . . . . . . . 9 (𝑋 ∈ (Base‘𝐾) → 𝐾 ∈ dom Base)
2 pleval2.b . . . . . . . . 9 𝐵 = (Base‘𝐾)
31, 2eleq2s 2821 . . . . . . . 8 (𝑋𝐵𝐾 ∈ dom Base)
43adantr 472 . . . . . . 7 ((𝑋𝐵𝑌𝐵) → 𝐾 ∈ dom Base)
5 pleval2.l . . . . . . . . 9 = (le‘𝐾)
6 pleval2.s . . . . . . . . 9 < = (lt‘𝐾)
75, 6pltval 17082 . . . . . . . 8 ((𝐾 ∈ dom Base ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ (𝑋 𝑌𝑋𝑌)))
873expb 1113 . . . . . . 7 ((𝐾 ∈ dom Base ∧ (𝑋𝐵𝑌𝐵)) → (𝑋 < 𝑌 ↔ (𝑋 𝑌𝑋𝑌)))
94, 8mpancom 706 . . . . . 6 ((𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ (𝑋 𝑌𝑋𝑌)))
109biimpar 503 . . . . 5 (((𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌𝑋𝑌)) → 𝑋 < 𝑌)
1110expr 644 . . . 4 (((𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) → (𝑋𝑌𝑋 < 𝑌))
1211necon1bd 2914 . . 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 1596  wcel 2103  wne 2896   class class class wbr 4760  dom cdm 5218  cfv 6001  Basecbs 15980  lecple 16071  ltcplt 17063
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-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-rab 3023  df-v 3306  df-sbc 3542  df-dif 3683  df-un 3685  df-in 3687  df-ss 3694  df-nul 4024  df-if 4195  df-sn 4286  df-pr 4288  df-op 4292  df-uni 4545  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-iota 5964  df-fun 6003  df-fv 6009  df-plt 17080
This theorem is referenced by:  pleval2  17087  pospo  17095
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