Theorem ple1 17044

Description: Any element is less than or equal to a poset's upper bound (if defined). (Contributed by NM, 22-Oct-2011.) (Revised by NM, 13-Sep-2018.)

Hypotheses

Ref	Expression
ple1.b	|- B = ( Base ` K )
ple1.u	|- U = ( lub ` K )
ple1.l	|- .<_ = ( le ` K )
ple1.1	|- .1. = ( 1. ` K )
ple1.k	|- ( ph -> K e. V )
ple1.x	|- ( ph -> X e. B )
ple1.d	|- ( ph -> B e. dom U )

Assertion

Ref	Expression
ple1	|- ( ph -> X .<_ .1. )

Proof of Theorem ple1

Step	Hyp	Ref	Expression
1		ple1.b	. . 3 |- B = ( Base ` K )
2		ple1.l	. . 3 |- .<_ = ( le ` K )
3		ple1.u	. . 3 |- U = ( lub ` K )
4		ple1.k	. . 3 |- ( ph -> K e. V )
5		ple1.d	. . 3 |- ( ph -> B e. dom U )
6		ple1.x	. . 3 |- ( ph -> X e. B )
7	1, 2, 3, 4, 5, 6	luble 16987	. 2 |- ( ph -> X .<_ ( U ` B ) )
8		ple1.1	. . . 4 |- .1. = ( 1. ` K )
9	1, 3, 8	p1val 17042	. . 3 |- ( K e. V -> .1. = ( U ` B ) )
10	4, 9	syl 17	. 2 |- ( ph -> .1. = ( U ` B ) )
11	7, 10	breqtrrd 4681	1 |- ( ph -> X .<_ .1. )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   class class class wbr 4653   dom cdm 5114   ` cfv 5888   Basecbs 15857   lecple 15948   lubclub 16942   1.cp1 17038

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-rep 4771  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-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  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-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-lub 16974  df-p1 17040

This theorem is referenced by:  ople1  34478  lhp2lt  35287