Theorem axltirr 8245

Description: Real number less-than is irreflexive. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltirr 8143 with ordering on the extended reals. New proofs should use ltnr 8255 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 8143	. 2 |- ( A e. RR -> -. A <RR A )
2		ltxrlt 8244	. . 3 |- ( ( A e. RR /\ A e. RR ) -> ( A < A <-> A <RR A ) )
3	2	anidms 397	. 2 |- ( A e. RR -> ( A < A <-> A <RR A ) )
4	1, 3	mtbird 679	1 |- ( A e. RR -> -. A < A )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105   e. wcel 2202   class class class wbr 4088   RRcr 8030   <RR cltrr 8035   < clt 8213

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-pre-ltirr 8143

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-xp 4731  df-pnf 8215  df-mnf 8216  df-ltxr 8218

This theorem is referenced by:  ltnr  8255