pnfnlt - Intuitionistic Logic Explorer

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Theorem pnfnlt 9744

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 7961	. . . . . . 7
2	1	neli 2437	. . . . . 6
3	2	intnanr 925	. . . . 5 $/\$
4	3	intnanr 925	. . . 4 $/\$ $/\$
5		pnfnemnf 7974	. . . . . 6
6	5	neii 2342	. . . . 5
7	6	intnanr 925	. . . 4 $/\$
8	4, 7	pm3.2ni 808	. . 3 $/\$ $/\$ $\/$ $/\$
9	2	intnanr 925	. . . 4 $/\$
10	6	intnanr 925	. . . 4 $/\$
11	9, 10	pm3.2ni 808	. . 3 $/\$ $\/$ $/\$
12	8, 11	pm3.2ni 808	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
13		pnfxr 7972	. . 3
14		ltxr 9732	. . 3 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
15	13, 14	mpan 422	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
16	12, 15	mtbiri 670	1

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103

wb 104 $\/$ wo 703

wceq 1348

wcel 2141 class class class wbr 3989

cr 7773

cltrr 7778

cpnf 7951

cmnf 7952

cxr 7953

clt 7954

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-cnex 7865 ax-resscn 7866

This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-opab 4051 df-xp 4617 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959

This theorem is referenced by: pnfge 9746 xrltnsym 9750 xrlttr 9752 xrltso 9753 xltnegi 9792 xposdif 9839 qbtwnxr 10214 xrmaxiflemab 11210 xrmaxltsup 11221