pnfnlt - Intuitionistic Logic Explorer

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Theorem pnfnlt 9723

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 7940	. . . . . . 7
2	1	neli 2433	. . . . . 6
3	2	intnanr 920	. . . . 5 $/\$
4	3	intnanr 920	. . . 4 $/\$ $/\$
5		pnfnemnf 7953	. . . . . 6
6	5	neii 2338	. . . . 5
7	6	intnanr 920	. . . 4 $/\$
8	4, 7	pm3.2ni 803	. . 3 $/\$ $/\$ $\/$ $/\$
9	2	intnanr 920	. . . 4 $/\$
10	6	intnanr 920	. . . 4 $/\$
11	9, 10	pm3.2ni 803	. . 3 $/\$ $\/$ $/\$
12	8, 11	pm3.2ni 803	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
13		pnfxr 7951	. . 3
14		ltxr 9711	. . 3 $/\$ $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
15	13, 14	mpan 421	. 2 $/\$ $/\$ $\/$ $/\$ $\/$ $/\$ $\/$ $/\$
16	12, 15	mtbiri 665	1

Colors of variables: wff set class

Syntax hints:

wn 3

wi 4 $/\$ wa 103

wb 104 $\/$ wo 698

wceq 1343

wcel 2136 class class class wbr 3982

cr 7752

cltrr 7757

cpnf 7930

cmnf 7931

cxr 7932

clt 7933

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 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-sep 4100 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-cnex 7844 ax-resscn 7845

This theorem depends on definitions: df-bi 116 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-nel 2432 df-ral 2449 df-rex 2450 df-rab 2453 df-v 2728 df-un 3120 df-in 3122 df-ss 3129 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-br 3983 df-opab 4044 df-xp 4610 df-pnf 7935 df-mnf 7936 df-xr 7937 df-ltxr 7938

This theorem is referenced by: pnfge 9725 xrltnsym 9729 xrlttr 9731 xrltso 9732 xltnegi 9771 xposdif 9818 qbtwnxr 10193 xrmaxiflemab 11188 xrmaxltsup 11199