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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 5541. 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 495 |
. . . 4
| |
| 2 | simplr 496 |
. . . 4
| |
| 3 | 1, 2 | addcld 7138 |
. . 3
|
| 4 | simprl 497 |
. . . 4
| |
| 5 | simprr 498 |
. . . 4
| |
| 6 | 4, 5 | addcld 7138 |
. . 3
|
| 7 | 4, 2 | addcld 7138 |
. . 3
|
| 8 | apcotr 7707 |
. . 3
# # #
| |
| 9 | 3, 6, 7, 8 | syl3anc 1169 |
. 2
#
#
#
|
| 10 | apadd1 7708 |
. . . 4
#
 #
| |
| 11 | 1, 4, 2, 10 | syl3anc 1169 |
. . 3
#
# |
| 12 | apadd2 7709 |
. . . . 5
#
#
| |
| 13 | 2, 5, 4, 12 | syl3anc 1169 |
. . . 4
#
# |
| 14 | apsym 7706 |
. . . . 5
# # | |
| 15 | 7, 6, 14 | syl2anc 403 |
. . . 4
#
# |
| 16 | 13, 15 | bitrd 186 |
. . 3
#
# |
| 17 | 11, 16 | orbi12d 739 |
. 2
#
# #
#
|
| 18 | 9, 17 | sylibrd 167 |
1
#
#
# |
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 102
wb 103 wo 661 wcel 1433 class class class wbr 3785
(class class class)co 5532 cc 6979 caddc 6984 # cap 7681 |
| This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1cn 7069 ax-1re 7070 ax-icn 7071 ax-addcl 7072 ax-addrcl 7073 ax-mulcl 7074 ax-mulrcl 7075 ax-addcom 7076 ax-mulcom 7077 ax-addass 7078 ax-mulass 7079 ax-distr 7080 ax-i2m1 7081 ax-0lt1 7082 ax-1rid 7083 ax-0id 7084 ax-rnegex 7085 ax-precex 7086 ax-cnre 7087 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 ax-pre-apti 7091 ax-pre-ltadd 7092 ax-pre-mulgt0 7093 |
| This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-reu 2355 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 df-riota 5488 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-pnf 7155 df-mnf 7156 df-ltxr 7158 df-sub 7281 df-neg 7282 df-reap 7675 df-ap 7682 |
| This theorem is referenced by: mulext1 7712 abs00ap 9948 absext 9949 |
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