Theorem pleval2 16289

Description: Less-than-or-equal in terms of less-than. (sspss 3518 analog.) (Contributed by NM, 17-Oct-2011.) (Revised 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
pleval2	|- ( ( K e. Poset /\ X e. B /\ Y e. B ) -> ( X .<_ Y <-> ( X .< Y \/ X = Y ) ) )

Proof of Theorem pleval2

Step	Hyp	Ref	Expression
1		pleval2.b	. . . 4 |- B = ( Base ` K )
2		pleval2.l	. . . 4 |- .<_ = ( le ` K )
3		pleval2.s	. . . 4 |- .< = ( lt ` K )
4	1, 2, 3	pleval2i 16288	. . 3 |- ( ( X e. B /\ Y e. B ) -> ( X .<_ Y -> ( X .< Y \/ X = Y ) ) )
5	4	3adant1 1048	. 2 |- ( ( K e. Poset /\ X e. B /\ Y e. B ) -> ( X .<_ Y -> ( X .< Y \/ X = Y ) ) )
6	2, 3	pltle 16285	. . 3 |- ( ( K e. Poset /\ X e. B /\ Y e. B ) -> ( X .< Y -> X .<_ Y ) )
7	1, 2	posref 16274	. . . . 5 |- ( ( K e. Poset /\ X e. B ) -> X .<_ X )
8	7	3adant3 1050	. . . 4 |- ( ( K e. Poset /\ X e. B /\ Y e. B ) -> X .<_ X )
9		breq2 4399	. . . 4 |- ( X = Y -> ( X .<_ X <-> X .<_ Y ) )
10	8, 9	syl5ibcom 228	. . 3 |- ( ( K e. Poset /\ X e. B /\ Y e. B ) -> ( X = Y -> X .<_ Y ) )
11	6, 10	jaod 387	. 2 |- ( ( K e. Poset /\ X e. B /\ Y e. B ) -> ( ( X .< Y \/ X = Y ) -> X .<_ Y ) )
12	5, 11	impbid 195	1 |- ( ( K e. Poset /\ X e. B /\ Y e. B ) -> ( X .<_ Y <-> ( X .< Y \/ X = Y ) ) )

Syntax hints:   -> wi 4   <-> wb 189   \/ wo 375   /\ w3a 1007   = wceq 1452   e. wcel 1904   class class class wbr 4395   ` cfv 5589   Basecbs 15199   lecple 15275   Posetcpo 16263   ltcplt 16264

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5553  df-fun 5591  df-fv 5597  df-preset 16251  df-poset 16269  df-plt 16282

This theorem is referenced by:  pltletr  16295  plelttr  16296  tosso  16360  tlt3  28501  orngsqr  28641