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Mirrors > Home > NFE Home > Th. List > imadif | Unicode version |
Description: The image of a difference is the difference of images. (Contributed by NM, 24-May-1998.) |
Ref | Expression |
---|---|
imadif |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | anandir 802 | . . . . . . 7 | |
2 | 1 | exbii 1582 | . . . . . 6 |
3 | 19.40 1609 | . . . . . 6 | |
4 | 2, 3 | sylbi 187 | . . . . 5 |
5 | nfv 1619 | . . . . . . . . . 10 | |
6 | nfe1 1732 | . . . . . . . . . 10 | |
7 | 5, 6 | nfan 1824 | . . . . . . . . 9 |
8 | funmo 5125 | . . . . . . . . . . . . 13 | |
9 | brcnv 4892 | . . . . . . . . . . . . . 14 | |
10 | 9 | mobii 2240 | . . . . . . . . . . . . 13 |
11 | 8, 10 | sylib 188 | . . . . . . . . . . . 12 |
12 | mopick 2266 | . . . . . . . . . . . 12 | |
13 | 11, 12 | sylan 457 | . . . . . . . . . . 11 |
14 | 13 | con2d 107 | . . . . . . . . . 10 |
15 | imnan 411 | . . . . . . . . . 10 | |
16 | 14, 15 | sylib 188 | . . . . . . . . 9 |
17 | 7, 16 | alrimi 1765 | . . . . . . . 8 |
18 | 17 | ex 423 | . . . . . . 7 |
19 | exancom 1586 | . . . . . . 7 | |
20 | alnex 1543 | . . . . . . 7 | |
21 | 18, 19, 20 | 3imtr3g 260 | . . . . . 6 |
22 | 21 | anim2d 548 | . . . . 5 |
23 | 4, 22 | syl5 28 | . . . 4 |
24 | 19.29r 1597 | . . . . . 6 | |
25 | 20, 24 | sylan2br 462 | . . . . 5 |
26 | andi 837 | . . . . . . 7 | |
27 | ianor 474 | . . . . . . . 8 | |
28 | 27 | anbi2i 675 | . . . . . . 7 |
29 | an32 773 | . . . . . . . 8 | |
30 | pm3.24 852 | . . . . . . . . . . 11 | |
31 | 30 | intnan 880 | . . . . . . . . . 10 |
32 | anass 630 | . . . . . . . . . 10 | |
33 | 31, 32 | mtbir 290 | . . . . . . . . 9 |
34 | 33 | biorfi 396 | . . . . . . . 8 |
35 | 29, 34 | bitri 240 | . . . . . . 7 |
36 | 26, 28, 35 | 3bitr4i 268 | . . . . . 6 |
37 | 36 | exbii 1582 | . . . . 5 |
38 | 25, 37 | sylib 188 | . . . 4 |
39 | 23, 38 | impbid1 194 | . . 3 |
40 | elima2 4755 | . . . 4 | |
41 | eldif 3221 | . . . . . 6 | |
42 | 41 | anbi1i 676 | . . . . 5 |
43 | 42 | exbii 1582 | . . . 4 |
44 | 40, 43 | bitri 240 | . . 3 |
45 | eldif 3221 | . . . 4 | |
46 | elima2 4755 | . . . . 5 | |
47 | elima2 4755 | . . . . . 6 | |
48 | 47 | notbii 287 | . . . . 5 |
49 | 46, 48 | anbi12i 678 | . . . 4 |
50 | 45, 49 | bitri 240 | . . 3 |
51 | 39, 44, 50 | 3bitr4g 279 | . 2 |
52 | 51 | eqrdv 2351 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wn 3 wi 4 wo 357 wa 358 wal 1540 wex 1541 wceq 1642 wcel 1710 wmo 2205 cdif 3206 class class class wbr 4639 cima 4722 ccnv 4771 wfun 4775 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-opab 4623 df-br 4640 df-co 4726 df-ima 4727 df-id 4767 df-cnv 4785 df-fun 4789 |
This theorem is referenced by: imain 5172 resdif 5306 |
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