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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 5776. 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 518 |
. . . 4
| |
| 2 | simplr 519 |
. . . 4
| |
| 3 | 1, 2 | addcld 7778 |
. . 3
|
| 4 | simprl 520 |
. . . 4
| |
| 5 | simprr 521 |
. . . 4
| |
| 6 | 4, 5 | addcld 7778 |
. . 3
|
| 7 | 4, 2 | addcld 7778 |
. . 3
|
| 8 | apcotr 8362 |
. . 3
# # #
| |
| 9 | 3, 6, 7, 8 | syl3anc 1216 |
. 2
#
#
#
|
| 10 | apadd1 8363 |
. . . 4
#
 #
| |
| 11 | 1, 4, 2, 10 | syl3anc 1216 |
. . 3
#
# |
| 12 | apadd2 8364 |
. . . . 5
#
#
| |
| 13 | 2, 5, 4, 12 | syl3anc 1216 |
. . . 4
#
# |
| 14 | apsym 8361 |
. . . . 5
# # | |
| 15 | 7, 6, 14 | syl2anc 408 |
. . . 4
#
# |
| 16 | 13, 15 | bitrd 187 |
. . 3
#
# |
| 17 | 11, 16 | orbi12d 782 |
. 2
#
# #
#
|
| 18 | 9, 17 | sylibrd 168 |
1
#
#
# |
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 103
wb 104 wo 697 wcel 1480 class class class wbr 3924
(class class class)co 5767 cc 7611 caddc 7616 # cap 8336 |
| 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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-precex 7723 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 ax-pre-mulgt0 7730 |
| This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-pnf 7795 df-mnf 7796 df-ltxr 7798 df-sub 7928 df-neg 7929 df-reap 8330 df-ap 8337 |
| This theorem is referenced by: mulext1 8367 abs00ap 10827 absext 10828 |
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