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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 5325, ax-un 7677. (Revised by BTernaryTau, 29-Nov-2024.) |
Ref | Expression |
---|---|
dom0 | β’ (π΄ βΌ β β π΄ = β ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brdomi 8905 | . . 3 β’ (π΄ βΌ β β βπ π:π΄β1-1ββ ) | |
2 | f1f 6743 | . . . . 5 β’ (π:π΄β1-1ββ β π:π΄βΆβ ) | |
3 | f00 6729 | . . . . . 6 β’ (π:π΄βΆβ β (π = β β§ π΄ = β )) | |
4 | 3 | simprbi 498 | . . . . 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 1934 | . . 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 8964 | . . 3 β’ (π΄ β β β π΄ = β ) | |
9 | endom 8926 | . . 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 1542 βwex 1782 β c0 4287 class class class wbr 5110 βΆwf 6497 β1-1βwf1 6498 β cen 8887 βΌ cdom 8888 |
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 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-br 5111 df-opab 5173 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-en 8891 df-dom 8892 |
This theorem is referenced by: sdom0 9059 0sdom1dom 9189 fin1a2lem11 10353 cfpwsdom 10527 |
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