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Theorem pleval2i 17569
Description: One direction of pleval2 17570. (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 6701 . . . . . . . . 9 (𝑋 ∈ (Base‘𝐾) → 𝐾 ∈ dom Base)
2 pleval2.b . . . . . . . . 9 𝐵 = (Base‘𝐾)
31, 2eleq2s 2936 . . . . . . . 8 (𝑋𝐵𝐾 ∈ dom Base)
43adantr 481 . . . . . . 7 ((𝑋𝐵𝑌𝐵) → 𝐾 ∈ dom Base)
5 pleval2.l . . . . . . . . 9 = (le‘𝐾)
6 pleval2.s . . . . . . . . 9 < = (lt‘𝐾)
75, 6pltval 17565 . . . . . . . 8 ((𝐾 ∈ dom Base ∧ 𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ (𝑋 𝑌𝑋𝑌)))
873expb 1114 . . . . . . 7 ((𝐾 ∈ dom Base ∧ (𝑋𝐵𝑌𝐵)) → (𝑋 < 𝑌 ↔ (𝑋 𝑌𝑋𝑌)))
94, 8mpancom 684 . . . . . 6 ((𝑋𝐵𝑌𝐵) → (𝑋 < 𝑌 ↔ (𝑋 𝑌𝑋𝑌)))
109biimpar 478 . . . . 5 (((𝑋𝐵𝑌𝐵) ∧ (𝑋 𝑌𝑋𝑌)) → 𝑋 < 𝑌)
1110expr 457 . . . 4 (((𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) → (𝑋𝑌𝑋 < 𝑌))
1211necon1bd 3039 . . 3 (((𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) → (¬ 𝑋 < 𝑌𝑋 = 𝑌))
1312orrd 859 . 2 (((𝑋𝐵𝑌𝐵) ∧ 𝑋 𝑌) → (𝑋 < 𝑌𝑋 = 𝑌))
1413ex 413 1 ((𝑋𝐵𝑌𝐵) → (𝑋 𝑌 → (𝑋 < 𝑌𝑋 = 𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wo 843   = wceq 1530  wcel 2107  wne 3021   class class class wbr 5063  dom cdm 5554  cfv 6354  Basecbs 16478  lecple 16567  ltcplt 17546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-iota 6313  df-fun 6356  df-fv 6362  df-plt 17563
This theorem is referenced by:  pleval2  17570  pospo  17578
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