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Mirrors > Home > ILE Home > Th. List > pnfxr | Unicode version |
Description: Plus infinity belongs to the set of extended reals. (Contributed by NM, 13-Oct-2005.) (Proof shortened by Anthony Hart, 29-Aug-2011.) |
Ref | Expression |
---|---|
pnfxr |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssun2 3210 | . . 3 | |
2 | df-pnf 7770 | . . . . 5 | |
3 | cnex 7712 | . . . . . . 7 | |
4 | 3 | uniex 4329 | . . . . . 6 |
5 | 4 | pwex 4077 | . . . . 5 |
6 | 2, 5 | eqeltri 2190 | . . . 4 |
7 | 6 | prid1 3599 | . . 3 |
8 | 1, 7 | sselii 3064 | . 2 |
9 | df-xr 7772 | . 2 | |
10 | 8, 9 | eleqtrri 2193 | 1 |
Colors of variables: wff set class |
Syntax hints: wcel 1465 cvv 2660 cun 3039 cpw 3480 cpr 3498 cuni 3706 cc 7586 cr 7587 cpnf 7765 cmnf 7766 cxr 7767 |
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-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-un 4325 ax-cnex 7679 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-uni 3707 df-pnf 7770 df-xr 7772 |
This theorem is referenced by: pnfex 7787 pnfnemnf 7788 xnn0xr 9013 xrltnr 9534 ltpnf 9535 mnfltpnf 9539 pnfnlt 9541 pnfge 9543 xrlttri3 9551 nltpnft 9565 xgepnf 9567 xrrebnd 9570 xrre 9571 xrre2 9572 xnegcl 9583 xaddf 9595 xaddval 9596 xaddpnf1 9597 xaddpnf2 9598 pnfaddmnf 9601 mnfaddpnf 9602 xrex 9607 xaddass2 9621 xltadd1 9627 xlt2add 9631 xsubge0 9632 xposdif 9633 xleaddadd 9638 elioc2 9687 elico2 9688 elicc2 9689 ioomax 9699 iccmax 9700 ioopos 9701 elioopnf 9718 elicopnf 9720 unirnioo 9724 elxrge0 9729 hashinfom 10492 rexico 10961 xrmaxiflemcl 10982 xrmaxadd 10998 xblpnfps 12494 xblpnf 12495 xblss2ps 12500 blssec 12534 blpnfctr 12535 reopnap 12634 blssioo 12641 |
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