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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 6022. 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 8192 |
. . 3
|
| 4 | simprl 529 |
. . . 4
| |
| 5 | simprr 531 |
. . . 4
| |
| 6 | 4, 5 | addcld 8192 |
. . 3
|
| 7 | 4, 2 | addcld 8192 |
. . 3
|
| 8 | apcotr 8780 |
. . 3
# # #
| |
| 9 | 3, 6, 7, 8 | syl3anc 1271 |
. 2
#
#
#
|
| 10 | apadd1 8781 |
. . . 4
#
 #
| |
| 11 | 1, 4, 2, 10 | syl3anc 1271 |
. . 3
#
# |
| 12 | apadd2 8782 |
. . . . 5
#
#
| |
| 13 | 2, 5, 4, 12 | syl3anc 1271 |
. . . 4
#
# |
| 14 | apsym 8779 |
. . . . 5
# # | |
| 15 | 7, 6, 14 | syl2anc 411 |
. . . 4
#
# |
| 16 | 13, 15 | bitrd 188 |
. . 3
#
# |
| 17 | 11, 16 | orbi12d 798 |
. 2
#
# #
#
|
| 18 | 9, 17 | sylibrd 169 |
1
#
#
# |
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 104
wb 105 wo 713 wcel 2200 class class class wbr 4086
(class class class)co 6013 cc 8023 caddc 8028 # cap 8754 |
| 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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-sep 4205 ax-pow 4262 ax-pr 4297 ax-un 4528 ax-setind 4633 ax-cnex 8116 ax-resscn 8117 ax-1cn 8118 ax-1re 8119 ax-icn 8120 ax-addcl 8121 ax-addrcl 8122 ax-mulcl 8123 ax-mulrcl 8124 ax-addcom 8125 ax-mulcom 8126 ax-addass 8127 ax-mulass 8128 ax-distr 8129 ax-i2m1 8130 ax-0lt1 8131 ax-1rid 8132 ax-0id 8133 ax-rnegex 8134 ax-precex 8135 ax-cnre 8136 ax-pre-ltirr 8137 ax-pre-ltwlin 8138 ax-pre-lttrn 8139 ax-pre-apti 8140 ax-pre-ltadd 8141 ax-pre-mulgt0 8142 |
| This theorem depends on definitions: df-bi 117 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rab 2517 df-v 2802 df-sbc 3030 df-dif 3200 df-un 3202 df-in 3204 df-ss 3211 df-pw 3652 df-sn 3673 df-pr 3674 df-op 3676 df-uni 3892 df-br 4087 df-opab 4149 df-id 4388 df-xp 4729 df-rel 4730 df-cnv 4731 df-co 4732 df-dm 4733 df-iota 5284 df-fun 5326 df-fv 5332 df-riota 5966 df-ov 6016 df-oprab 6017 df-mpo 6018 df-pnf 8209 df-mnf 8210 df-ltxr 8212 df-sub 8345 df-neg 8346 df-reap 8748 df-ap 8755 |
| This theorem is referenced by: mulext1 8785 abs00ap 11616 absext 11617 |
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