ltxr - Intuitionistic Logic Explorer

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Theorem ltxr 10156

Description: The 'less than' binary relation on the set of extended reals. Definition 12-3.1 of [Gleason] p. 173. (Contributed by NM, 14-Oct-2005.)

**Assertion**
Ref	Expression
ltxr	$/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$

Proof of Theorem ltxr Dummy variables

are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq12 4130	. . . . 5 $/\$
2		df-3an 1011	. . . . . 6 $/\$ $/\$ $/\$ $/\$
3	2	opabbii 4193	. . . . 5 $/\$ $/\$ $/\$ $/\$
4	1, 3	brab2ga 4845	. . . 4 $/\$ $/\$ $/\$ $/\$
5	4	a1i 9	. . 3 $/\$ $/\$ $/\$ $/\$ $/\$
6		brun 4177	. . . 4 $\/$
7		brxp 4800	. . . . . . 7 $/\$
8		elun 3370	. . . . . . . . . . 11 $\/$
9		orcom 740	. . . . . . . . . . 11 $\/$ $\/$
10	8, 9	bitri 184	. . . . . . . . . 10 $\/$
11		elsng 3720	. . . . . . . . . . 11
12	11	orbi1d 803	. . . . . . . . . 10 $\/$ $\/$
13	10, 12	bitrid 192	. . . . . . . . 9 $\/$
14		elsng 3720	. . . . . . . . 9
15	13, 14	bi2anan9 614	. . . . . . . 8 $/\$ $/\$ $\/$ $/\$
16		andir 831	. . . . . . . 8 $\/$ $/\$ $/\$ $\/$ $/\$
17	15, 16	bitrdi 196	. . . . . . 7 $/\$ $/\$ $/\$ $\/$ $/\$
18	7, 17	bitrid 192	. . . . . 6 $/\$ $/\$ $\/$ $/\$
19		brxp 4800	. . . . . . 7 $/\$
20	11	anbi1d 469	. . . . . . . 8 $/\$ $/\$
21	20	adantr 276	. . . . . . 7 $/\$ $/\$ $/\$
22	19, 21	bitrid 192	. . . . . 6 $/\$ $/\$
23	18, 22	orbi12d 805	. . . . 5 $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
24		orass 779	. . . . 5 $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$
25	23, 24	bitrdi 196	. . . 4 $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
26	6, 25	bitrid 192	. . 3 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$
27	5, 26	orbi12d 805	. 2 $/\$ $/\$ $/\$ $\/$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
28		df-ltxr 8355	. . . 4 $/\$ $/\$
29	28	breqi 4131	. . 3 $/\$ $/\$
30		brun 4177	. . 3 $/\$ $/\$ $/\$ $/\$ $\/$
31	29, 30	bitri 184	. 2 $/\$ $/\$ $\/$
32		orass 779	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
33	27, 31, 32	3bitr4g 223	1 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$

Colors of variables: wff set class

Syntax hints:

wi 4 $/\$ wa 104

wb 105 $\/$ wo 720 $/\$ w3a 1009

wceq 1402

wcel 2209

cun 3218

csn 3705 class class class wbr 4125

copab 4186

cxp 4767

cr 8168

cltrr 8173

cpnf 8347

cmnf 8348

cxr 8349

clt 8350

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-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4244 ax-pow 4306 ax-pr 4341

This theorem depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ral 2533 df-rex 2534 df-v 2823 df-un 3224 df-in 3226 df-ss 3233 df-pw 3687 df-sn 3711 df-pr 3712 df-op 3714 df-br 4126 df-opab 4188 df-xp 4775 df-ltxr 8355

This theorem is referenced by: xrltnr 10160 ltpnf 10161 mnflt 10164 mnfltpnf 10166 pnfnlt 10168 nltmnf 10169

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