Theorem pleval2 16204

Description: Less-than-or-equal in terms of less-than. (sspss 3531 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 16203	. . 3 |- ( ( X e. B /\ Y e. B ) -> ( X .<_ Y -> ( X .< Y \/ X = Y ) ) )
5	4	3adant1 1025	. 2 |- ( ( K e. Poset /\ X e. B /\ Y e. B ) -> ( X .<_ Y -> ( X .< Y \/ X = Y ) ) )
6	2, 3	pltle 16200	. . 3 |- ( ( K e. Poset /\ X e. B /\ Y e. B ) -> ( X .< Y -> X .<_ Y ) )
7	1, 2	posref 16189	. . . . 5 |- ( ( K e. Poset /\ X e. B ) -> X .<_ X )
8	7	3adant3 1027	. . . 4 |- ( ( K e. Poset /\ X e. B /\ Y e. B ) -> X .<_ X )
9		breq2 4405	. . . 4 |- ( X = Y -> ( X .<_ X <-> X .<_ Y ) )
10	8, 9	syl5ibcom 224	. . 3 |- ( ( K e. Poset /\ X e. B /\ Y e. B ) -> ( X = Y -> X .<_ Y ) )
11	6, 10	jaod 382	. 2 |- ( ( K e. Poset /\ X e. B /\ Y e. B ) -> ( ( X .< Y \/ X = Y ) -> X .<_ Y ) )
12	5, 11	impbid 194	1 |- ( ( K e. Poset /\ X e. B /\ Y e. B ) -> ( X .<_ Y <-> ( X .< Y \/ X = Y ) ) )

Syntax hints:   -> wi 4   <-> wb 188   \/ wo 370   /\ w3a 984   = wceq 1443   e. wcel 1886   class class class wbr 4401   ` cfv 5581   Basecbs 15114   lecple 15190   Posetcpo 16178   ltcplt 16179

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-sbc 3267  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-iota 5545  df-fun 5583  df-fv 5589  df-preset 16166  df-poset 16184  df-plt 16197

This theorem is referenced by:  pltletr  16210  plelttr  16211  tosso  16275  tlt3  28419  orngsqr  28560