![]() |
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 12823 | . . 3 β’ β€β₯:β€βΆπ« β€ | |
2 | ffn 6708 | . . 3 β’ (β€β₯:β€βΆπ« β€ β β€β₯ Fn β€) | |
3 | fvelrnb 6943 | . . 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 12835 | . . . . 5 β’ (π β β€ β π β (β€β₯βπ)) | |
6 | 5 | ne0d 4328 | . . . 4 β’ (π β β€ β (β€β₯βπ) β β ) |
7 | neeq1 2995 | . . . 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 3140 | . 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 1533 β wcel 2098 β wne 2932 βwrex 3062 β c0 4315 π« cpw 4595 ran crn 5668 Fn wfn 6529 βΆwf 6530 βcfv 6534 β€cz 12556 β€β₯cuz 12820 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pow 5354 ax-pr 5418 ax-un 7719 ax-cnex 11163 ax-resscn 11164 ax-pre-lttri 11181 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-f1 6539 df-fo 6540 df-f1o 6541 df-fv 6542 df-ov 7405 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11248 df-mnf 11249 df-xr 11250 df-ltxr 11251 df-le 11252 df-neg 11445 df-z 12557 df-uz 12821 |
This theorem is referenced by: heibor1lem 37171 |
Copyright terms: Public domain | W3C validator |