pnfnlt - Intuitionistic Logic Explorer

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Theorem pnfnlt 9853

Description: No extended real is greater than plus infinity. (Contributed by NM, 15-Oct-2005.)

**Assertion**
Ref	Expression
pnfnlt

**Proof of Theorem pnfnlt**
Step	Hyp	Ref	Expression
1		pnfnre 8061	. . . . . . 7
2	1	neli 2461	. . . . . 6
3	2	intnanr 931	. . . . 5 $/\$
4	3	intnanr 931	. . . 4 $/\$ $/\$
5		pnfnemnf 8074	. . . . . 6
6	5	neii 2366	. . . . 5
7	6	intnanr 931	. . . 4 $/\$
8	4, 7	pm3.2ni 814	. . 3 $/\$ $/\$ $\/$ $/\$
9	2	intnanr 931	. . . 4 $/\$
10	6	intnanr 931	. . . 4 $/\$
11	9, 10	pm3.2ni 814	. . 3 $/\$ $\/$ $/\$
12	8, 11	pm3.2ni 814	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
13		pnfxr 8072	. . 3
14		ltxr 9841	. . 3 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
15	13, 14	mpan 424	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
16	12, 15	mtbiri 676	1

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 709

wceq 1364

wcel 2164 class class class wbr 4029

cr 7871

cltrr 7876

cpnf 8051

cmnf 8052

cxr 8053

clt 8054

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 4147 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-cnex 7963 ax-resscn 7964

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-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-br 4030 df-opab 4091 df-xp 4665 df-pnf 8056 df-mnf 8057 df-xr 8058 df-ltxr 8059

This theorem is referenced by: pnfge 9855 xrltnsym 9859 xrlttr 9861 xrltso 9862 xltnegi 9901 xposdif 9948 qbtwnxr 10326 xqltnle 10336 xrmaxiflemab 11390 xrmaxltsup 11401