Theorem axltirr 8154

Description: Real number less-than is irreflexive. Axiom for real and complex numbers, derived from set theory. This restates ax-pre-ltirr 8052 with ordering on the extended reals. New proofs should use ltnr 8164 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 8052	. 2 |- ( A e. RR -> -. A <RR A )
2		ltxrlt 8153	. . 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 675	1 |- ( A e. RR -> -. A < A )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105   e. wcel 2177   class class class wbr 4050   RRcr 7939   <RR cltrr 7944   < clt 8122

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4169  ax-pow 4225  ax-pr 4260  ax-un 4487  ax-setind 4592  ax-cnex 8031  ax-resscn 8032  ax-pre-ltirr 8052

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-rab 2494  df-v 2775  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-pw 3622  df-sn 3643  df-pr 3644  df-op 3646  df-uni 3856  df-br 4051  df-opab 4113  df-xp 4688  df-pnf 8124  df-mnf 8125  df-ltxr 8127

This theorem is referenced by:  ltnr  8164