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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 5833. 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 7897 |
. . 3
|
| 4 | simprl 521 |
. . . 4
| |
| 5 | simprr 522 |
. . . 4
| |
| 6 | 4, 5 | addcld 7897 |
. . 3
|
| 7 | 4, 2 | addcld 7897 |
. . 3
|
| 8 | apcotr 8482 |
. . 3
# # #
| |
| 9 | 3, 6, 7, 8 | syl3anc 1220 |
. 2
#
#
#
|
| 10 | apadd1 8483 |
. . . 4
#
 #
| |
| 11 | 1, 4, 2, 10 | syl3anc 1220 |
. . 3
#
# |
| 12 | apadd2 8484 |
. . . . 5
#
#
| |
| 13 | 2, 5, 4, 12 | syl3anc 1220 |
. . . 4
#
# |
| 14 | apsym 8481 |
. . . . 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 2128 class class class wbr 3965
(class class class)co 5824 cc 7730 caddc 7735 # cap 8456 |
| 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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-cnex 7823 ax-resscn 7824 ax-1cn 7825 ax-1re 7826 ax-icn 7827 ax-addcl 7828 ax-addrcl 7829 ax-mulcl 7830 ax-mulrcl 7831 ax-addcom 7832 ax-mulcom 7833 ax-addass 7834 ax-mulass 7835 ax-distr 7836 ax-i2m1 7837 ax-0lt1 7838 ax-1rid 7839 ax-0id 7840 ax-rnegex 7841 ax-precex 7842 ax-cnre 7843 ax-pre-ltirr 7844 ax-pre-ltwlin 7845 ax-pre-lttrn 7846 ax-pre-apti 7847 ax-pre-ltadd 7848 ax-pre-mulgt0 7849 |
| This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-br 3966 df-opab 4026 df-id 4253 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-iota 5135 df-fun 5172 df-fv 5178 df-riota 5780 df-ov 5827 df-oprab 5828 df-mpo 5829 df-pnf 7914 df-mnf 7915 df-ltxr 7917 df-sub 8048 df-neg 8049 df-reap 8450 df-ap 8457 |
| This theorem is referenced by: mulext1 8487 abs00ap 10962 absext 10963 |
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