Nearby theorems
|
|
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
| 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 5791. 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 7809 |
. . 3
|
| 4 | simprl 521 |
. . . 4
| |
| 5 | simprr 522 |
. . . 4
| |
| 6 | 4, 5 | addcld 7809 |
. . 3
|
| 7 | 4, 2 | addcld 7809 |
. . 3
|
| 8 | apcotr 8393 |
. . 3
# # #
| |
| 9 | 3, 6, 7, 8 | syl3anc 1217 |
. 2
#
#
#
|
| 10 | apadd1 8394 |
. . . 4
#
 #
| |
| 11 | 1, 4, 2, 10 | syl3anc 1217 |
. . 3
#
# |
| 12 | apadd2 8395 |
. . . . 5
#
#
| |
| 13 | 2, 5, 4, 12 | syl3anc 1217 |
. . . 4
#
# |
| 14 | apsym 8392 |
. . . . 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 1481 class class class wbr 3937
(class class class)co 5782 cc 7642 caddc 7647 # cap 8367 |
| 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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-mulrcl 7743 ax-addcom 7744 ax-mulcom 7745 ax-addass 7746 ax-mulass 7747 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-1rid 7751 ax-0id 7752 ax-rnegex 7753 ax-precex 7754 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-apti 7759 ax-pre-ltadd 7760 ax-pre-mulgt0 7761 |
| This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-iota 5096 df-fun 5133 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-pnf 7826 df-mnf 7827 df-ltxr 7829 df-sub 7959 df-neg 7960 df-reap 8361 df-ap 8368 |
| This theorem is referenced by: mulext1 8398 abs00ap 10866 absext 10867 |
| Copyright terms: Public domain | W3C validator |