Theorem reapirr 8598

Description: Real apartness is irreflexive. Part of Definition 11.2.7(v) of [HoTT], p. (varies). Beyond the development of # itself, proofs should use apirr 8626 instead. (Contributed by Jim Kingdon, 26-Jan-2020.)

Assertion

Ref	Expression
reapirr	|- ( A e. RR -> -. A #ℝ A )

Proof of Theorem reapirr

Step	Hyp	Ref	Expression
1		ltnr 8098	. 2 |- ( A e. RR -> -. A < A )
2		reapval 8597	. . . 4 |- ( ( A e. RR /\ A e. RR ) -> ( A #ℝ A <-> ( A < A \/ A < A ) ) )
3	2	anidms 397	. . 3 |- ( A e. RR -> ( A #ℝ A <-> ( A < A \/ A < A ) ) )
4		oridm 758	. . 3 |- ( ( A < A \/ A < A ) <-> A < A )
5	3, 4	bitrdi 196	. 2 |- ( A e. RR -> ( A #ℝ A <-> A < A ) )
6	1, 5	mtbird 674	1 |- ( A e. RR -> -. A #ℝ A )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105   \/ wo 709   e. wcel 2164   class class class wbr 4030   RRcr 7873   < clt 8056   #_ℝ creap 8595

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-pre-ltirr 7986

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-xp 4666  df-pnf 8058  df-mnf 8059  df-ltxr 8061  df-reap 8596

This theorem is referenced by:  apirr  8626