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Mirrors > Home > ILE Home > Th. List > dif1enen | Unicode version |
Description: Subtracting one element from each of two equinumerous finite sets. (Contributed by Jim Kingdon, 5-Jun-2022.) |
Ref | Expression |
---|---|
dif1enen.a |
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dif1enen.ab |
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dif1enen.c |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
dif1enen.d |
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Ref | Expression |
---|---|
dif1enen |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dif1enen.a |
. . 3
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2 | isfi 6760 |
. . 3
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3 | 1, 2 | sylib 122 |
. 2
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4 | simplrr 536 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
5 | breq2 4007 |
. . . . . . 7
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6 | 5 | adantl 277 |
. . . . . 6
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7 | 4, 6 | mpbid 147 |
. . . . 5
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8 | en0 6794 |
. . . . 5
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9 | 7, 8 | sylib 122 |
. . . 4
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10 | dif1enen.c |
. . . . . 6
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11 | n0i 3428 |
. . . . . 6
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12 | 10, 11 | syl 14 |
. . . . 5
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13 | 12 | ad2antrr 488 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
14 | 9, 13 | pm2.21dd 620 |
. . 3
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15 | simplr 528 |
. . . . . . . 8
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16 | simprr 531 |
. . . . . . . . . 10
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17 | 16 | ad2antrr 488 |
. . . . . . . . 9
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18 | breq2 4007 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
19 | 18 | adantl 277 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
20 | 17, 19 | mpbid 147 |
. . . . . . . 8
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21 | 10 | ad3antrrr 492 |
. . . . . . . 8
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22 | dif1en 6878 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
23 | 15, 20, 21, 22 | syl3anc 1238 |
. . . . . . 7
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24 | dif1enen.ab |
. . . . . . . . . . . 12
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
25 | 24 | ad3antrrr 492 |
. . . . . . . . . . 11
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26 | 25 | ensymd 6782 |
. . . . . . . . . 10
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27 | entr 6783 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
28 | 26, 20, 27 | syl2anc 411 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
29 | dif1enen.d |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
30 | 29 | ad3antrrr 492 |
. . . . . . . . 9
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31 | dif1en 6878 |
. . . . . . . . 9
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
32 | 15, 28, 30, 31 | syl3anc 1238 |
. . . . . . . 8
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33 | 32 | ensymd 6782 |
. . . . . . 7
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34 | entr 6783 |
. . . . . . 7
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35 | 23, 33, 34 | syl2anc 411 |
. . . . . 6
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36 | 35 | ex 115 |
. . . . 5
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37 | 36 | rexlimdva 2594 |
. . . 4
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38 | 37 | imp 124 |
. . 3
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39 | nn0suc 4603 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
40 | 39 | ad2antrl 490 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
41 | 14, 38, 40 | mpjaodan 798 |
. 2
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42 | 3, 41 | rexlimddv 2599 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4118 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-iinf 4587 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-tr 4102 df-id 4293 df-iord 4366 df-on 4368 df-suc 4371 df-iom 4590 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-er 6534 df-en 6740 df-fin 6742 |
This theorem is referenced by: fisseneq 6930 |
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