pnfnlt - Intuitionistic Logic Explorer

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Theorem pnfnlt 9911

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 8116	. . . . . . 7
2	1	neli 2473	. . . . . 6
3	2	intnanr 932	. . . . 5 $/\$
4	3	intnanr 932	. . . 4 $/\$ $/\$
5		pnfnemnf 8129	. . . . . 6
6	5	neii 2378	. . . . 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 8127	. . 3
14		ltxr 9899	. . 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 2176 class class class wbr 4045

cr 7926

cltrr 7931

cpnf 8106

cmnf 8107

cxr 8108

clt 8109

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 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-sep 4163 ax-pow 4219 ax-pr 4254 ax-un 4481 ax-cnex 8018 ax-resscn 8019

This theorem depends on definitions: df-bi 117 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-nel 2472 df-ral 2489 df-rex 2490 df-rab 2493 df-v 2774 df-un 3170 df-in 3172 df-ss 3179 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-br 4046 df-opab 4107 df-xp 4682 df-pnf 8111 df-mnf 8112 df-xr 8113 df-ltxr 8114

This theorem is referenced by: pnfge 9913 xrltnsym 9917 xrlttr 9919 xrltso 9920 xltnegi 9959 xposdif 10006 qbtwnxr 10402 xqltnle 10412 xrmaxiflemab 11591 xrmaxltsup 11602