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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 5934. 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 528 |
. . . 4
| |
| 3 | 1, 2 | addcld 8065 |
. . 3
|
| 4 | simprl 529 |
. . . 4
| |
| 5 | simprr 531 |
. . . 4
| |
| 6 | 4, 5 | addcld 8065 |
. . 3
|
| 7 | 4, 2 | addcld 8065 |
. . 3
|
| 8 | apcotr 8653 |
. . 3
# # #
| |
| 9 | 3, 6, 7, 8 | syl3anc 1249 |
. 2
#
#
#
|
| 10 | apadd1 8654 |
. . . 4
#
 #
| |
| 11 | 1, 4, 2, 10 | syl3anc 1249 |
. . 3
#
# |
| 12 | apadd2 8655 |
. . . . 5
#
#
| |
| 13 | 2, 5, 4, 12 | syl3anc 1249 |
. . . 4
#
# |
| 14 | apsym 8652 |
. . . . 5
# # | |
| 15 | 7, 6, 14 | syl2anc 411 |
. . . 4
#
# |
| 16 | 13, 15 | bitrd 188 |
. . 3
#
# |
| 17 | 11, 16 | orbi12d 794 |
. 2
#
# #
#
|
| 18 | 9, 17 | sylibrd 169 |
1
#
#
# |
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 104
wb 105 wo 709 wcel 2167 class class class wbr 4034
(class class class)co 5925 cc 7896 caddc 7901 # cap 8627 |
| 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 615 ax-in2 616 ax-io 710 ax-5 1461 ax-7 1462 ax-gen 1463 ax-ie1 1507 ax-ie2 1508 ax-8 1518 ax-10 1519 ax-11 1520 ax-i12 1521 ax-bndl 1523 ax-4 1524 ax-17 1540 ax-i9 1544 ax-ial 1548 ax-i5r 1549 ax-13 2169 ax-14 2170 ax-ext 2178 ax-sep 4152 ax-pow 4208 ax-pr 4243 ax-un 4469 ax-setind 4574 ax-cnex 7989 ax-resscn 7990 ax-1cn 7991 ax-1re 7992 ax-icn 7993 ax-addcl 7994 ax-addrcl 7995 ax-mulcl 7996 ax-mulrcl 7997 ax-addcom 7998 ax-mulcom 7999 ax-addass 8000 ax-mulass 8001 ax-distr 8002 ax-i2m1 8003 ax-0lt1 8004 ax-1rid 8005 ax-0id 8006 ax-rnegex 8007 ax-precex 8008 ax-cnre 8009 ax-pre-ltirr 8010 ax-pre-ltwlin 8011 ax-pre-lttrn 8012 ax-pre-apti 8013 ax-pre-ltadd 8014 ax-pre-mulgt0 8015 |
| This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1475 df-sb 1777 df-eu 2048 df-mo 2049 df-clab 2183 df-cleq 2189 df-clel 2192 df-nfc 2328 df-ne 2368 df-nel 2463 df-ral 2480 df-rex 2481 df-reu 2482 df-rab 2484 df-v 2765 df-sbc 2990 df-dif 3159 df-un 3161 df-in 3163 df-ss 3170 df-pw 3608 df-sn 3629 df-pr 3630 df-op 3632 df-uni 3841 df-br 4035 df-opab 4096 df-id 4329 df-xp 4670 df-rel 4671 df-cnv 4672 df-co 4673 df-dm 4674 df-iota 5220 df-fun 5261 df-fv 5267 df-riota 5880 df-ov 5928 df-oprab 5929 df-mpo 5930 df-pnf 8082 df-mnf 8083 df-ltxr 8085 df-sub 8218 df-neg 8219 df-reap 8621 df-ap 8628 |
| This theorem is referenced by: mulext1 8658 abs00ap 11246 absext 11247 |
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