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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 6059. 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 527 |
. . . 4
| |
| 2 | simplr 529 |
. . . 4
| |
| 3 | 1, 2 | addcld 8293 |
. . 3
|
| 4 | simprl 531 |
. . . 4
| |
| 5 | simprr 533 |
. . . 4
| |
| 6 | 4, 5 | addcld 8293 |
. . 3
|
| 7 | 4, 2 | addcld 8293 |
. . 3
|
| 8 | apcotr 8881 |
. . 3
# # #
| |
| 9 | 3, 6, 7, 8 | syl3anc 1274 |
. 2
#
#
#
|
| 10 | apadd1 8882 |
. . . 4
#
 #
| |
| 11 | 1, 4, 2, 10 | syl3anc 1274 |
. . 3
#
# |
| 12 | apadd2 8883 |
. . . . 5
#
#
| |
| 13 | 2, 5, 4, 12 | syl3anc 1274 |
. . . 4
#
# |
| 14 | apsym 8880 |
. . . . 5
# # | |
| 15 | 7, 6, 14 | syl2anc 411 |
. . . 4
#
# |
| 16 | 13, 15 | bitrd 188 |
. . 3
#
# |
| 17 | 11, 16 | orbi12d 801 |
. 2
#
# #
#
|
| 18 | 9, 17 | sylibrd 169 |
1
#
#
# |
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 104
wb 105 wo 716 wcel 2203 class class class wbr 4109
(class class class)co 6050 cc 8125 caddc 8130 # cap 8855 |
| 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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-sep 4228 ax-pow 4287 ax-pr 4322 ax-un 4554 ax-setind 4659 ax-cnex 8218 ax-resscn 8219 ax-1cn 8220 ax-1re 8221 ax-icn 8222 ax-addcl 8223 ax-addrcl 8224 ax-mulcl 8225 ax-mulrcl 8226 ax-addcom 8227 ax-mulcom 8228 ax-addass 8229 ax-mulass 8230 ax-distr 8231 ax-i2m1 8232 ax-0lt1 8233 ax-1rid 8234 ax-0id 8235 ax-rnegex 8236 ax-precex 8237 ax-cnre 8238 ax-pre-ltirr 8239 ax-pre-ltwlin 8240 ax-pre-lttrn 8241 ax-pre-apti 8242 ax-pre-ltadd 8243 ax-pre-mulgt0 8244 |
| This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ne 2413 df-nel 2508 df-ral 2525 df-rex 2526 df-reu 2527 df-rab 2529 df-v 2815 df-sbc 3043 df-dif 3213 df-un 3215 df-in 3217 df-ss 3224 df-pw 3671 df-sn 3695 df-pr 3696 df-op 3698 df-uni 3915 df-br 4110 df-opab 4172 df-id 4414 df-xp 4755 df-rel 4756 df-cnv 4757 df-co 4758 df-dm 4759 df-iota 5312 df-fun 5354 df-fv 5360 df-riota 6003 df-ov 6053 df-oprab 6054 df-mpo 6055 df-pnf 8310 df-mnf 8311 df-ltxr 8313 df-sub 8446 df-neg 8447 df-reap 8849 df-ap 8856 |
| This theorem is referenced by: mulext1 8886 abs00ap 11747 absext 11748 |
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