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Mirrors > Home > MPE Home > Th. List > rn0 | Structured version Visualization version Unicode version |
Description: The range of the empty set is empty. Part of Theorem 3.8(v) of [Monk1] p. 36. (Contributed by NM, 4-Jul-1994.) |
Ref | Expression |
---|---|
rn0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dm0 5339 | . 2 | |
2 | dm0rn0 5342 | . 2 | |
3 | 1, 2 | mpbi 220 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wceq 1483 c0 3915 cdm 5114 crn 5115 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-cnv 5122 df-dm 5124 df-rn 5125 |
This theorem is referenced by: ima0 5481 0ima 5482 rnxpid 5567 xpima 5576 f0 6086 2ndval 7171 frxp 7287 oarec 7642 fodomr 8111 dfac5lem3 8948 itunitc 9243 0rest 16090 arwval 16693 pmtrfrn 17878 psgnsn 17940 oppglsm 18057 mpfrcl 19518 ply1frcl 19683 edgval 25941 0grsubgr 26170 0uhgrsubgr 26171 0ngrp 27365 bafval 27459 locfinref 29908 esumrnmpt2 30130 sibf0 30396 mvtval 31397 mrsubrn 31410 mrsub0 31413 mrsubf 31414 mrsubccat 31415 mrsubcn 31416 mrsubco 31418 mrsubvrs 31419 elmsubrn 31425 msubrn 31426 msubf 31429 mstaval 31441 mzpmfp 37310 dmnonrel 37896 imanonrel 37899 conrel1d 37955 clsneibex 38400 neicvgbex 38410 sge00 40593 |
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