pnfnlt - Intuitionistic Logic Explorer

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Theorem pnfnlt 9944

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 8149	. . . . . . 7
2	1	neli 2475	. . . . . 6
3	2	intnanr 932	. . . . 5 $/\$
4	3	intnanr 932	. . . 4 $/\$ $/\$
5		pnfnemnf 8162	. . . . . 6
6	5	neii 2380	. . . . 5
7	6	intnanr 932	. . . 4 $/\$
8	4, 7	pm3.2ni 815	. . 3 $/\$ $/\$ $\/$ $/\$
9	2	intnanr 932	. . . 4 $/\$
10	6	intnanr 932	. . . 4 $/\$
11	9, 10	pm3.2ni 815	. . 3 $/\$ $\/$ $/\$
12	8, 11	pm3.2ni 815	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
13		pnfxr 8160	. . 3
14		ltxr 9932	. . 3 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
15	13, 14	mpan 424	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
16	12, 15	mtbiri 677	1

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 104

wb 105 $\/$ wo 710

wceq 1373

wcel 2178 class class class wbr 4059

cr 7959

cltrr 7964

cpnf 8139

cmnf 8140

cxr 8141

clt 8142

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 2180 ax-14 2181 ax-ext 2189 ax-sep 4178 ax-pow 4234 ax-pr 4269 ax-un 4498 ax-cnex 8051 ax-resscn 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 2194 df-cleq 2200 df-clel 2203 df-nfc 2339 df-ne 2379 df-nel 2474 df-ral 2491 df-rex 2492 df-rab 2495 df-v 2778 df-un 3178 df-in 3180 df-ss 3187 df-pw 3628 df-sn 3649 df-pr 3650 df-op 3652 df-uni 3865 df-br 4060 df-opab 4122 df-xp 4699 df-pnf 8144 df-mnf 8145 df-xr 8146 df-ltxr 8147

This theorem is referenced by: pnfge 9946 xrltnsym 9950 xrlttr 9952 xrltso 9953 xltnegi 9992 xposdif 10039 qbtwnxr 10437 xqltnle 10447 xrmaxiflemab 11673 xrmaxltsup 11684