Theorem reapirr 8747

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 8775 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 8246	. 2 |- ( A e. RR -> -. A < A )
2		reapval 8746	. . . 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 762	. . 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 677	1 |- ( A e. RR -> -. A #ℝ A )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105   \/ wo 713   e. wcel 2200   class class class wbr 4086   RRcr 8021   < clt 8204   #_ℝ creap 8744

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-cnex 8113  ax-resscn 8114  ax-pre-ltirr 8134

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2802  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-opab 4149  df-xp 4729  df-pnf 8206  df-mnf 8207  df-ltxr 8209  df-reap 8745

This theorem is referenced by:  apirr  8775