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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 7950 | . . . . 5 | |
2 | 1 | neli 2437 | . . . 4 |
3 | nn0re 9133 | . . . 4 | |
4 | 2, 3 | mto 657 | . . 3 |
5 | eleq1 2233 | . . 3 | |
6 | 4, 5 | mtbiri 670 | . 2 |
7 | 6 | necon2ai 2394 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wceq 1348 wcel 2141 wne 2340 cr 7762 cpnf 7940 cn0 9124 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-un 4416 ax-cnex 7854 ax-resscn 7855 ax-1re 7857 ax-addrcl 7860 ax-rnegex 7872 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-uni 3795 df-int 3830 df-pnf 7945 df-inn 8868 df-n0 9125 |
This theorem is referenced by: nn0nepnfd 9197 fxnn0nninf 10383 0tonninf 10384 1tonninf 10385 |
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