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| Mirrors > Home > ILE Home > Th. List > addext | Unicode version | ||
| Description: Strong extensionality for addition. Given excluded middle, apartness would be equivalent to negated equality and this would follow readily (for all operations) from oveq12 5851. For us, it is proved a different way. (Contributed by Jim Kingdon, 15-Feb-2020.) |
| Ref | Expression |
|---|---|
| addext |
     ![/\](wedge.gif "/\"){kind=small}       #
#
 # |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simpll 519 |
. . . 4
| |
| 2 | simplr 520 |
. . . 4
| |
| 3 | 1, 2 | addcld 7918 |
. . 3
|
| 4 | simprl 521 |
. . . 4
| |
| 5 | simprr 522 |
. . . 4
| |
| 6 | 4, 5 | addcld 7918 |
. . 3
|
| 7 | 4, 2 | addcld 7918 |
. . 3
|
| 8 | apcotr 8505 |
. . 3
# # #
| |
| 9 | 3, 6, 7, 8 | syl3anc 1228 |
. 2
#
#
#
|
| 10 | apadd1 8506 |
. . . 4
#
 #
| |
| 11 | 1, 4, 2, 10 | syl3anc 1228 |
. . 3
#
# |
| 12 | apadd2 8507 |
. . . . 5
#
#
| |
| 13 | 2, 5, 4, 12 | syl3anc 1228 |
. . . 4
#
# |
| 14 | apsym 8504 |
. . . . 5
# # | |
| 15 | 7, 6, 14 | syl2anc 409 |
. . . 4
#
# |
| 16 | 13, 15 | bitrd 187 |
. . 3
#
# |
| 17 | 11, 16 | orbi12d 783 |
. 2
#
# #
#
|
| 18 | 9, 17 | sylibrd 168 |
1
#
#
# |
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 103
wb 104 wo 698 wcel 2136 class class class wbr 3982
(class class class)co 5842 cc 7751 caddc 7756 # cap 8479 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1435 ax-7 1436 ax-gen 1437 ax-ie1 1481 ax-ie2 1482 ax-8 1492 ax-10 1493 ax-11 1494 ax-i12 1495 ax-bndl 1497 ax-4 1498 ax-17 1514 ax-i9 1518 ax-ial 1522 ax-i5r 1523 ax-13 2138 ax-14 2139 ax-ext 2147 ax-sep 4100 ax-pow 4153 ax-pr 4187 ax-un 4411 ax-setind 4514 ax-cnex 7844 ax-resscn 7845 ax-1cn 7846 ax-1re 7847 ax-icn 7848 ax-addcl 7849 ax-addrcl 7850 ax-mulcl 7851 ax-mulrcl 7852 ax-addcom 7853 ax-mulcom 7854 ax-addass 7855 ax-mulass 7856 ax-distr 7857 ax-i2m1 7858 ax-0lt1 7859 ax-1rid 7860 ax-0id 7861 ax-rnegex 7862 ax-precex 7863 ax-cnre 7864 ax-pre-ltirr 7865 ax-pre-ltwlin 7866 ax-pre-lttrn 7867 ax-pre-apti 7868 ax-pre-ltadd 7869 ax-pre-mulgt0 7870 |
| This theorem depends on definitions: df-bi 116 df-3an 970 df-tru 1346 df-fal 1349 df-nf 1449 df-sb 1751 df-eu 2017 df-mo 2018 df-clab 2152 df-cleq 2158 df-clel 2161 df-nfc 2297 df-ne 2337 df-nel 2432 df-ral 2449 df-rex 2450 df-reu 2451 df-rab 2453 df-v 2728 df-sbc 2952 df-dif 3118 df-un 3120 df-in 3122 df-ss 3129 df-pw 3561 df-sn 3582 df-pr 3583 df-op 3585 df-uni 3790 df-br 3983 df-opab 4044 df-id 4271 df-xp 4610 df-rel 4611 df-cnv 4612 df-co 4613 df-dm 4614 df-iota 5153 df-fun 5190 df-fv 5196 df-riota 5798 df-ov 5845 df-oprab 5846 df-mpo 5847 df-pnf 7935 df-mnf 7936 df-ltxr 7938 df-sub 8071 df-neg 8072 df-reap 8473 df-ap 8480 |
| This theorem is referenced by: mulext1 8510 abs00ap 11004 absext 11005 |
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