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Mirrors > Home > ILE Home > Th. List > nn0nepnf | Unicode version |
Description: No standard nonnegative integer equals positive infinity. (Contributed by AV, 10-Dec-2020.) |
Ref | Expression |
---|---|
nn0nepnf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pnfnre 7800 | . . . . 5 | |
2 | 1 | neli 2403 | . . . 4 |
3 | nn0re 8979 | . . . 4 | |
4 | 2, 3 | mto 651 | . . 3 |
5 | eleq1 2200 | . . 3 | |
6 | 4, 5 | mtbiri 664 | . 2 |
7 | 6 | necon2ai 2360 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1331 wcel 1480 wne 2306 cr 7612 cpnf 7790 cn0 8970 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-un 4350 ax-cnex 7704 ax-resscn 7705 ax-1re 7707 ax-addrcl 7710 ax-rnegex 7722 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-uni 3732 df-int 3767 df-pnf 7795 df-inn 8714 df-n0 8971 |
This theorem is referenced by: nn0nepnfd 9043 fxnn0nninf 10204 0tonninf 10205 1tonninf 10206 |
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