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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 6027. 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 529 |
. . . 4
| |
| 3 | 1, 2 | addcld 8199 |
. . 3
|
| 4 | simprl 531 |
. . . 4
| |
| 5 | simprr 533 |
. . . 4
| |
| 6 | 4, 5 | addcld 8199 |
. . 3
|
| 7 | 4, 2 | addcld 8199 |
. . 3
|
| 8 | apcotr 8787 |
. . 3
# # #
| |
| 9 | 3, 6, 7, 8 | syl3anc 1273 |
. 2
#
#
#
|
| 10 | apadd1 8788 |
. . . 4
#
 #
| |
| 11 | 1, 4, 2, 10 | syl3anc 1273 |
. . 3
#
# |
| 12 | apadd2 8789 |
. . . . 5
#
#
| |
| 13 | 2, 5, 4, 12 | syl3anc 1273 |
. . . 4
#
# |
| 14 | apsym 8786 |
. . . . 5
# # | |
| 15 | 7, 6, 14 | syl2anc 411 |
. . . 4
#
# |
| 16 | 13, 15 | bitrd 188 |
. . 3
#
# |
| 17 | 11, 16 | orbi12d 800 |
. 2
#
# #
#
|
| 18 | 9, 17 | sylibrd 169 |
1
#
#
# |
| Colors of variables: wff set class |
| Syntax hints: wi 4
wa 104
wb 105 wo 715 wcel 2202 class class class wbr 4088
(class class class)co 6018 cc 8030 caddc 8035 # cap 8761 |
| 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 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2204 ax-14 2205 ax-ext 2213 ax-sep 4207 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-cnex 8123 ax-resscn 8124 ax-1cn 8125 ax-1re 8126 ax-icn 8127 ax-addcl 8128 ax-addrcl 8129 ax-mulcl 8130 ax-mulrcl 8131 ax-addcom 8132 ax-mulcom 8133 ax-addass 8134 ax-mulass 8135 ax-distr 8136 ax-i2m1 8137 ax-0lt1 8138 ax-1rid 8139 ax-0id 8140 ax-rnegex 8141 ax-precex 8142 ax-cnre 8143 ax-pre-ltirr 8144 ax-pre-ltwlin 8145 ax-pre-lttrn 8146 ax-pre-apti 8147 ax-pre-ltadd 8148 ax-pre-mulgt0 8149 |
| This theorem depends on definitions: df-bi 117 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rab 2519 df-v 2804 df-sbc 3032 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-br 4089 df-opab 4151 df-id 4390 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-iota 5286 df-fun 5328 df-fv 5334 df-riota 5971 df-ov 6021 df-oprab 6022 df-mpo 6023 df-pnf 8216 df-mnf 8217 df-ltxr 8219 df-sub 8352 df-neg 8353 df-reap 8755 df-ap 8762 |
| This theorem is referenced by: mulext1 8792 abs00ap 11624 absext 11625 |
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