Theorem meetcom 16831

Description: The meet of a poset commutes. (The antecedent (XMY) e. B /\ (YMX) e. B i.e. "the meets exist" could be omitted as an artefact of our particular join definition, but other definitions may require it.)

Hypotheses

Ref	Expression
meetcom.b	|- B = (base` K)
meetcom.m	|- M = (meet` K)

Assertion

Ref	Expression
meetcom	|- (((K e. PosetNEW /\ X e. B /\ Y e. B) /\ ((XMY) e. B /\ (YMX) e. B)) -> (XMY) = (YMX))

Proof of Theorem meetcom

Step	Hyp	Ref	Expression
1		meetcom.b	. . 3 |- B = (base` K)
2		meetcom.m	. . 3 |- M = (meet` K)
3	1, 2	meetcomALT 16830	. 2 |- ((K e. PosetNEW /\ X e. B /\ Y e. B) -> (XMY) = (YMX))
4	3	adantr 425	1 |- (((K e. PosetNEW /\ X e. B /\ Y e. B) /\ ((XMY) e. B /\ (YMX) e. B)) -> (XMY) = (YMX))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  ` cfv 3998  (class class class)co 4884  basecbs 16758  PosetNEWcpo 16760  meetcmee 16767

This theorem is referenced by:  latmcom 16870

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-mpt2 5007  df-meet 16802

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