pnfnlt - Intuitionistic Logic Explorer

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Theorem pnfnlt 9862

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 8068	. . . . . . 7
2	1	neli 2464	. . . . . 6
3	2	intnanr 931	. . . . 5 $/\$
4	3	intnanr 931	. . . 4 $/\$ $/\$
5		pnfnemnf 8081	. . . . . 6
6	5	neii 2369	. . . . 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 8079	. . 3
14		ltxr 9850	. . 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 2167 class class class wbr 4033

cr 7878

cltrr 7883

cpnf 8058

cmnf 8059

cxr 8060

clt 8061

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 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-sep 4151 ax-pow 4207 ax-pr 4242 ax-un 4468 ax-cnex 7970 ax-resscn 7971

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-rab 2484 df-v 2765 df-un 3161 df-in 3163 df-ss 3170 df-pw 3607 df-sn 3628 df-pr 3629 df-op 3631 df-uni 3840 df-br 4034 df-opab 4095 df-xp 4669 df-pnf 8063 df-mnf 8064 df-xr 8065 df-ltxr 8066

This theorem is referenced by: pnfge 9864 xrltnsym 9868 xrlttr 9870 xrltso 9871 xltnegi 9910 xposdif 9957 qbtwnxr 10347 xqltnle 10357 xrmaxiflemab 11412 xrmaxltsup 11423