![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > co02 | Structured version Visualization version Unicode version |
Description: Composition with the empty set. Theorem 20 of [Suppes] p. 63. (Contributed by NM, 24-Apr-2004.) |
Ref | Expression |
---|---|
co02 |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relco 5332 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | rel0 4957 |
. 2
![]() ![]() ![]() | |
3 | br0 4448 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() | |
4 | 3 | intnanr 925 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
5 | 4 | nex 1677 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
6 | vex 3047 |
. . . . 5
![]() ![]() ![]() ![]() | |
7 | vex 3047 |
. . . . 5
![]() ![]() ![]() ![]() | |
8 | 6, 7 | opelco 5005 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
9 | 5, 8 | mtbir 301 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
10 | noel 3734 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
11 | 9, 10 | 2false 352 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
12 | 1, 2, 11 | eqrelriiv 4928 |
1
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1668 ax-4 1681 ax-5 1757 ax-6 1804 ax-7 1850 ax-9 1895 ax-10 1914 ax-11 1919 ax-12 1932 ax-13 2090 ax-ext 2430 ax-sep 4524 ax-nul 4533 ax-pr 4638 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3an 986 df-tru 1446 df-ex 1663 df-nf 1667 df-sb 1797 df-eu 2302 df-mo 2303 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2580 df-ne 2623 df-ral 2741 df-rex 2742 df-rab 2745 df-v 3046 df-dif 3406 df-un 3408 df-in 3410 df-ss 3417 df-nul 3731 df-if 3881 df-sn 3968 df-pr 3970 df-op 3974 df-br 4402 df-opab 4461 df-xp 4839 df-rel 4840 df-co 4842 |
This theorem is referenced by: co01 5349 gsumwmhm 16622 frmdgsum 16639 frmdup1 16641 efginvrel2 17370 0frgp 17422 evl1fval 18909 ust0 21227 utop2nei 21258 tngds 21649 mrsub0 30147 dfpo2 30388 cononrel1 36194 |
Copyright terms: Public domain | W3C validator |