Theorem axltirr 7298

Description: Real number less-than is irreflexive. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltirr 7202 with ordering on the extended reals. New proofs should use ltnr 7307 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 7202	. 2 |- ( A e. RR -> -. A <RR A )
2		ltxrlt 7297	. . 3 |- ( ( A e. RR /\ A e. RR ) -> ( A < A <-> A <RR A ) )
3	2	anidms 389	. 2 |- ( A e. RR -> ( A < A <-> A <RR A ) )
4	1, 3	mtbird 631	1 |- ( A e. RR -> -. A < A )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 103   e. wcel 1434   class class class wbr 3805   RRcr 7094   <RR cltrr 7099   < clt 7267

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-cnex 7181  ax-resscn 7182  ax-pre-ltirr 7202

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2612  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-xp 4397  df-pnf 7269  df-mnf 7270  df-ltxr 7272

This theorem is referenced by:  ltnr  7307