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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 5677. 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 497 |
. . . 4
| |
| 2 | simplr 498 |
. . . 4
| |
| 3 | 1, 2 | addcld 7570 |
. . 3
|
| 4 | simprl 499 |
. . . 4
| |
| 5 | simprr 500 |
. . . 4
| |
| 6 | 4, 5 | addcld 7570 |
. . 3
|
| 7 | 4, 2 | addcld 7570 |
. . 3
|
| 8 | apcotr 8147 |
. . 3
# # #
| |
| 9 | 3, 6, 7, 8 | syl3anc 1175 |
. 2
#
#
#
|
| 10 | apadd1 8148 |
. . . 4
#
 #
| |
| 11 | 1, 4, 2, 10 | syl3anc 1175 |
. . 3
#
# |
| 12 | apadd2 8149 |
. . . . 5
#
#
| |
| 13 | 2, 5, 4, 12 | syl3anc 1175 |
. . . 4
#
# |
| 14 | apsym 8146 |
. . . . 5
# # | |
| 15 | 7, 6, 14 | syl2anc 404 |
. . . 4
#
# |
| 16 | 13, 15 | bitrd 187 |
. . 3
#
# |
| 17 | 11, 16 | orbi12d 743 |
. 2
#
# #
#
|
| 18 | 9, 17 | sylibrd 168 |
1
#
#
# |
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 103
wb 104 wo 665 wcel 1439 class class class wbr 3853
(class class class)co 5668 cc 7411 caddc 7416 # cap 8121 |
| This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3965 ax-pow 4017 ax-pr 4047 ax-un 4271 ax-setind 4368 ax-cnex 7499 ax-resscn 7500 ax-1cn 7501 ax-1re 7502 ax-icn 7503 ax-addcl 7504 ax-addrcl 7505 ax-mulcl 7506 ax-mulrcl 7507 ax-addcom 7508 ax-mulcom 7509 ax-addass 7510 ax-mulass 7511 ax-distr 7512 ax-i2m1 7513 ax-0lt1 7514 ax-1rid 7515 ax-0id 7516 ax-rnegex 7517 ax-precex 7518 ax-cnre 7519 ax-pre-ltirr 7520 ax-pre-ltwlin 7521 ax-pre-lttrn 7522 ax-pre-apti 7523 ax-pre-ltadd 7524 ax-pre-mulgt0 7525 |
| This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-reu 2367 df-rab 2369 df-v 2624 df-sbc 2844 df-dif 3004 df-un 3006 df-in 3008 df-ss 3015 df-pw 3437 df-sn 3458 df-pr 3459 df-op 3461 df-uni 3662 df-br 3854 df-opab 3908 df-id 4131 df-xp 4460 df-rel 4461 df-cnv 4462 df-co 4463 df-dm 4464 df-iota 4995 df-fun 5032 df-fv 5038 df-riota 5624 df-ov 5671 df-oprab 5672 df-mpt2 5673 df-pnf 7587 df-mnf 7588 df-ltxr 7590 df-sub 7718 df-neg 7719 df-reap 8115 df-ap 8122 |
| This theorem is referenced by: mulext1 8152 abs00ap 10558 absext 10559 |
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