![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > dom0 | Structured version Visualization version GIF version |
Description: A set dominated by the empty set is empty. (Contributed by NM, 22-Nov-2004.) Avoid ax-pow 5365, ax-un 7740. (Revised by BTernaryTau, 29-Nov-2024.) |
Ref | Expression |
---|---|
dom0 | β’ (π΄ βΌ β β π΄ = β ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brdomi 8978 | . . 3 β’ (π΄ βΌ β β βπ π:π΄β1-1ββ ) | |
2 | f1f 6793 | . . . . 5 β’ (π:π΄β1-1ββ β π:π΄βΆβ ) | |
3 | f00 6779 | . . . . . 6 β’ (π:π΄βΆβ β (π = β β§ π΄ = β )) | |
4 | 3 | simprbi 496 | . . . . 5 β’ (π:π΄βΆβ β π΄ = β ) |
5 | 2, 4 | syl 17 | . . . 4 β’ (π:π΄β1-1ββ β π΄ = β ) |
6 | 5 | exlimiv 1926 | . . 3 β’ (βπ π:π΄β1-1ββ β π΄ = β ) |
7 | 1, 6 | syl 17 | . 2 β’ (π΄ βΌ β β π΄ = β ) |
8 | en0 9037 | . . 3 β’ (π΄ β β β π΄ = β ) | |
9 | endom 8999 | . . 3 β’ (π΄ β β β π΄ βΌ β ) | |
10 | 8, 9 | sylbir 234 | . 2 β’ (π΄ = β β π΄ βΌ β ) |
11 | 7, 10 | impbii 208 | 1 β’ (π΄ βΌ β β π΄ = β ) |
Colors of variables: wff setvar class |
Syntax hints: β wb 205 = wceq 1534 βwex 1774 β c0 4323 class class class wbr 5148 βΆwf 6544 β1-1βwf1 6545 β cen 8960 βΌ cdom 8961 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5149 df-opab 5211 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-en 8964 df-dom 8965 |
This theorem is referenced by: sdom0 9132 0sdom1dom 9262 fin1a2lem11 10433 cfpwsdom 10607 |
Copyright terms: Public domain | W3C validator |