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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 6084. 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 531 |
. . . 4
| |
| 2 | simplr 533 |
. . . 4
| |
| 3 | 1, 2 | addcld 8335 |
. . 3
|
| 4 | simprl 535 |
. . . 4
| |
| 5 | simprr 537 |
. . . 4
| |
| 6 | 4, 5 | addcld 8335 |
. . 3
|
| 7 | 4, 2 | addcld 8335 |
. . 3
|
| 8 | apcotr 8925 |
. . 3
# # #
| |
| 9 | 3, 6, 7, 8 | syl3anc 1278 |
. 2
#
#
#
|
| 10 | apadd1 8926 |
. . . 4
#
 #
| |
| 11 | 1, 4, 2, 10 | syl3anc 1278 |
. . 3
#
# |
| 12 | apadd2 8927 |
. . . . 5
#
#
| |
| 13 | 2, 5, 4, 12 | syl3anc 1278 |
. . . 4
#
# |
| 14 | apsym 8924 |
. . . . 5
# # | |
| 15 | 7, 6, 14 | syl2anc 415 |
. . . 4
#
# |
| 16 | 13, 15 | bitrd 188 |
. . 3
#
# |
| 17 | 11, 16 | orbi12d 805 |
. 2
#
# #
#
|
| 18 | 9, 17 | sylibrd 169 |
1
#
#
# |
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 104
wb 105 wo 720 wcel 2209 class class class wbr 4125
(class class class)co 6075 cc 8167 caddc 8172 # cap 8899 |
| 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 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4244 ax-pow 4306 ax-pr 4341 ax-un 4573 ax-setind 4679 ax-cnex 8260 ax-resscn 8261 ax-1cn 8262 ax-1re 8263 ax-icn 8264 ax-addcl 8265 ax-addrcl 8266 ax-mulcl 8267 ax-mulrcl 8268 ax-addcom 8269 ax-mulcom 8270 ax-addass 8271 ax-mulass 8272 ax-distr 8273 ax-i2m1 8274 ax-0lt1 8275 ax-1rid 8276 ax-0id 8277 ax-rnegex 8278 ax-precex 8279 ax-cnre 8280 ax-pre-ltirr 8281 ax-pre-ltwlin 8282 ax-pre-lttrn 8283 ax-pre-apti 8284 ax-pre-ltadd 8285 ax-pre-mulgt0 8286 |
| This theorem depends on definitions: df-bi 117 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-reu 2535 df-rab 2537 df-v 2823 df-sbc 3052 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-pw 3687 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-br 4126 df-opab 4188 df-id 4433 df-xp 4775 df-rel 4776 df-cnv 4777 df-co 4778 df-dm 4779 df-iota 5332 df-fun 5374 df-fv 5380 df-riota 6028 df-ov 6078 df-oprab 6079 df-mpo 6080 df-pnf 8352 df-mnf 8353 df-ltxr 8355 df-sub 8489 df-neg 8490 df-reap 8893 df-ap 8900 |
| This theorem is referenced by: mulext1 8930 abs00ap 11806 absext 11807 |
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