Theorem pltirr 16963

Description: The less-than relation is not reflexive. (pssirr 3707 analog.) (Contributed by NM, 7-Feb-2012.)

Hypothesis

Ref	Expression
pltne.s	|- .< = ( lt ` K )

Assertion

Ref	Expression
pltirr	|- ( ( K e. A /\ X e. B ) -> -. X .< X )

Proof of Theorem pltirr

Step	Hyp	Ref	Expression
1		eqid 2622	. 2 |- X = X
2		pltne.s	. . . . 5 |- .< = ( lt ` K )
3	2	pltne 16962	. . . 4 |- ( ( K e. A /\ X e. B /\ X e. B ) -> ( X .< X -> X =/= X ) )
4	3	3anidm23 1385	. . 3 |- ( ( K e. A /\ X e. B ) -> ( X .< X -> X =/= X ) )
5	4	necon2bd 2810	. 2 |- ( ( K e. A /\ X e. B ) -> ( X = X -> -. X .< X ) )
6	1, 5	mpi 20	1 |- ( ( K e. A /\ X e. B ) -> -. X .< X )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   ` cfv 5888   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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  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:  pospo  16973  atnlt  34600  llnnlt  34809  lplnnlt  34851  lvolnltN  34904