![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > uzn0 | Structured version Visualization version GIF version |
Description: The upper integers are all nonempty. (Contributed by Mario Carneiro, 16-Jan-2014.) |
Ref | Expression |
---|---|
uzn0 | β’ (π β ran β€β₯ β π β β ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uzf 12774 | . . 3 β’ β€β₯:β€βΆπ« β€ | |
2 | ffn 6672 | . . 3 β’ (β€β₯:β€βΆπ« β€ β β€β₯ Fn β€) | |
3 | fvelrnb 6907 | . . 3 β’ (β€β₯ Fn β€ β (π β ran β€β₯ β βπ β β€ (β€β₯βπ) = π)) | |
4 | 1, 2, 3 | mp2b 10 | . 2 β’ (π β ran β€β₯ β βπ β β€ (β€β₯βπ) = π) |
5 | uzid 12786 | . . . . 5 β’ (π β β€ β π β (β€β₯βπ)) | |
6 | 5 | ne0d 4299 | . . . 4 β’ (π β β€ β (β€β₯βπ) β β ) |
7 | neeq1 3003 | . . . 4 β’ ((β€β₯βπ) = π β ((β€β₯βπ) β β β π β β )) | |
8 | 6, 7 | syl5ibcom 244 | . . 3 β’ (π β β€ β ((β€β₯βπ) = π β π β β )) |
9 | 8 | rexlimiv 3142 | . 2 β’ (βπ β β€ (β€β₯βπ) = π β π β β ) |
10 | 4, 9 | sylbi 216 | 1 β’ (π β ran β€β₯ β π β β ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 = wceq 1542 β wcel 2107 β wne 2940 βwrex 3070 β c0 4286 π« cpw 4564 ran crn 5638 Fn wfn 6495 βΆwf 6496 βcfv 6500 β€cz 12507 β€β₯cuz 12771 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-pre-lttri 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7364 df-er 8654 df-en 8890 df-dom 8891 df-sdom 8892 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-neg 11396 df-z 12508 df-uz 12772 |
This theorem is referenced by: heibor1lem 36318 |
Copyright terms: Public domain | W3C validator |