Theorem axltirr 7824

Description: Real number less-than is irreflexive. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltirr 7725 with ordering on the extended reals. New proofs should use ltnr 7834 instead for naming consistency. (New usage is discouraged.) (Contributed by Jim Kingdon, 15-Jan-2020.)

Assertion

Ref	Expression
axltirr	|- ( A e. RR -> -. A < A )

Proof of Theorem axltirr

Step	Hyp	Ref	Expression
1		ax-pre-ltirr 7725	. 2 |- ( A e. RR -> -. A <RR A )
2		ltxrlt 7823	. . 3 |- ( ( A e. RR /\ A e. RR ) -> ( A < A <-> A <RR A ) )
3	2	anidms 394	. 2 |- ( A e. RR -> ( A < A <-> A <RR A ) )
4	1, 3	mtbird 662	1 |- ( A e. RR -> -. A < A )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 104   e. wcel 1480   class class class wbr 3924   RRcr 7612   <RR cltrr 7617   < clt 7793

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705  ax-pre-ltirr 7725

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-xp 4540  df-pnf 7795  df-mnf 7796  df-ltxr 7798

This theorem is referenced by:  ltnr  7834