Theorem pleval2i 16964

Description: One direction of pleval2 16965. (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	|- ( ( X e. B /\ Y e. B ) -> ( X .<_ Y -> ( X .< Y \/ X = Y ) ) )

Proof of Theorem pleval2i

Step	Hyp	Ref	Expression
1		elfvdm 6220	. . . . . . . . 9 |- ( X e. ( Base ` K ) -> K e. dom Base )
2		pleval2.b	. . . . . . . . 9 |- B = ( Base ` K )
3	1, 2	eleq2s 2719	. . . . . . . 8 |- ( X e. B -> K e. dom Base )
4	3	adantr 481	. . . . . . 7 |- ( ( X e. B /\ Y e. B ) -> K e. dom Base )
5		pleval2.l	. . . . . . . . 9 |- .<_ = ( le ` K )
6		pleval2.s	. . . . . . . . 9 |- .< = ( lt ` K )
7	5, 6	pltval 16960	. . . . . . . 8 |- ( ( K e. dom Base /\ X e. B /\ Y e. B ) -> ( X .< Y <-> ( X .<_ Y /\ X =/= Y ) ) )
8	7	3expb 1266	. . . . . . 7 |- ( ( K e. dom Base /\ ( X e. B /\ Y e. B ) ) -> ( X .< Y <-> ( X .<_ Y /\ X =/= Y ) ) )
9	4, 8	mpancom 703	. . . . . 6 |- ( ( X e. B /\ Y e. B ) -> ( X .< Y <-> ( X .<_ Y /\ X =/= Y ) ) )
10	9	biimpar 502	. . . . 5 |- ( ( ( X e. B /\ Y e. B ) /\ ( X .<_ Y /\ X =/= Y ) ) -> X .< Y )
11	10	expr 643	. . . 4 |- ( ( ( X e. B /\ Y e. B ) /\ X .<_ Y ) -> ( X =/= Y -> X .< Y ) )
12	11	necon1bd 2812	. . 3 |- ( ( ( X e. B /\ Y e. B ) /\ X .<_ Y ) -> ( -. X .< Y -> X = Y ) )
13	12	orrd 393	. 2 |- ( ( ( X e. B /\ Y e. B ) /\ X .<_ Y ) -> ( X .< Y \/ X = Y ) )
14	13	ex 450	1 |- ( ( X e. B /\ Y e. B ) -> ( X .<_ Y -> ( X .< Y \/ X = Y ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   dom cdm 5114   ` cfv 5888   Basecbs 15857   lecple 15948   ltcplt 16941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-plt 16958

This theorem is referenced by:  pleval2  16965  pospo  16973