xqltnle - Intuitionistic Logic Explorer

	Intuitionistic Logic Explorer		< Previous Next > Nearby theorems
Mirrors > Home > ILE Home > Th. List > xqltnle		Unicode version

Theorem xqltnle 10357

Description: "Less than" expressed in terms of "less than or equal to", for extended numbers which are rational or

. We have not yet had enough usage of such numbers to warrant fully developing the concept, as in NN0* or

, so for now we just have a handful of theorems for what we need. (Contributed by Jim Kingdon, 5-Jun-2025.)

**Assertion**
Ref	Expression
xqltnle	$\/$ $/\$ $\/$

**Proof of Theorem xqltnle**
Step	Hyp	Ref	Expression
1		qltnle 10333	. . . 4 $/\$
2	1	adantll 476	. . 3 $\/$ $/\$ $\/$ $/\$ $/\$
3		simplr 528	. . . . . . 7 $\/$ $/\$ $\/$ $/\$ $/\$
4		qre 9699	. . . . . . 7
5	3, 4	syl 14	. . . . . 6 $\/$ $/\$ $\/$ $/\$ $/\$
6	5	ltpnfd 9856	. . . . 5 $\/$ $/\$ $\/$ $/\$ $/\$
7		simpr 110	. . . . 5 $\/$ $/\$ $\/$ $/\$ $/\$
8	6, 7	breqtrrd 4061	. . . 4 $\/$ $/\$ $\/$ $/\$ $/\$
9	5	renepnfd 8077	. . . . . . 7 $\/$ $/\$ $\/$ $/\$ $/\$
10	9	neneqd 2388	. . . . . 6 $\/$ $/\$ $\/$ $/\$ $/\$
11	5	rexrd 8076	. . . . . . 7 $\/$ $/\$ $\/$ $/\$ $/\$
12		xgepnf 9891	. . . . . . 7
13	11, 12	syl 14	. . . . . 6 $\/$ $/\$ $\/$ $/\$ $/\$
14	10, 13	mtbird 674	. . . . 5 $\/$ $/\$ $\/$ $/\$ $/\$
15	7, 14	eqnbrtrd 4051	. . . 4 $\/$ $/\$ $\/$ $/\$ $/\$
16	8, 15	2thd 175	. . 3 $\/$ $/\$ $\/$ $/\$ $/\$
17		simplr 528	. . 3 $\/$ $/\$ $\/$ $/\$ $\/$
18	2, 16, 17	mpjaodan 799	. 2 $\/$ $/\$ $\/$ $/\$
19		simpr 110	. . . 4 $\/$ $/\$ $\/$ $/\$
20		qre 9699	. . . . . . . . 9
21	20	rexrd 8076	. . . . . . . 8
22		pnfxr 8079	. . . . . . . . 9
23		eleq1 2259	. . . . . . . . 9
24	22, 23	mpbiri 168	. . . . . . . 8
25	21, 24	jaoi 717	. . . . . . 7 $\/$
26	25	adantl 277	. . . . . 6 $\/$ $/\$ $\/$
27		pnfnlt 9862	. . . . . 6
28	26, 27	syl 14	. . . . 5 $\/$ $/\$ $\/$
29	28	adantr 276	. . . 4 $\/$ $/\$ $\/$ $/\$
30	19, 29	eqnbrtrd 4051	. . 3 $\/$ $/\$ $\/$ $/\$
31		pnfge 9864	. . . . . . 7
32	26, 31	syl 14	. . . . . 6 $\/$ $/\$ $\/$
33	32	adantr 276	. . . . 5 $\/$ $/\$ $\/$ $/\$
34	33, 19	breqtrrd 4061	. . . 4 $\/$ $/\$ $\/$ $/\$
35	34	notnotd 631	. . 3 $\/$ $/\$ $\/$ $/\$
36	30, 35	2falsed 703	. 2 $\/$ $/\$ $\/$ $/\$
37		simpl 109	. 2 $\/$ $/\$ $\/$ $\/$
38	18, 36, 37	mpjaodan 799	1 $\/$ $/\$ $\/$

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 709

wceq 1364

wcel 2167 class class class wbr 4033

cr 7878

cpnf 8058

cxr 8060

clt 8061

cle 8062

cq 9693

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 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-sep 4151 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-setind 4573 ax-cnex 7970 ax-resscn 7971 ax-1cn 7972 ax-1re 7973 ax-icn 7974 ax-addcl 7975 ax-addrcl 7976 ax-mulcl 7977 ax-mulrcl 7978 ax-addcom 7979 ax-mulcom 7980 ax-addass 7981 ax-mulass 7982 ax-distr 7983 ax-i2m1 7984 ax-0lt1 7985 ax-1rid 7986 ax-0id 7987 ax-rnegex 7988 ax-precex 7989 ax-cnre 7990 ax-pre-ltirr 7991 ax-pre-ltwlin 7992 ax-pre-lttrn 7993 ax-pre-apti 7994 ax-pre-ltadd 7995 ax-pre-mulgt0 7996 ax-pre-mulext 7997

This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rmo 2483 df-rab 2484 df-v 2765 df-sbc 2990 df-csb 3085 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-int 3875 df-iun 3918 df-br 4034 df-opab 4095 df-mpt 4096 df-id 4328 df-po 4331 df-iso 4332 df-xp 4669 df-rel 4670 df-cnv 4671 df-co 4672 df-dm 4673 df-rn 4674 df-res 4675 df-ima 4676 df-iota 5219 df-fun 5260 df-fn 5261 df-f 5262 df-fv 5266 df-riota 5877 df-ov 5925 df-oprab 5926 df-mpo 5927 df-1st 6198 df-2nd 6199 df-pnf 8063 df-mnf 8064 df-xr 8065 df-ltxr 8066 df-le 8067 df-sub 8199 df-neg 8200 df-reap 8602 df-ap 8609 df-div 8700 df-inn 8991 df-n0 9250 df-z 9327 df-q 9694 df-rp 9729

This theorem is referenced by: pcadd2 12510

W3C validator