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| Mirrors > Home > NFE Home > Th. List > addcid1 | Unicode version | ||
| Description: Cardinal zero is a fixed point for cardinal addition. Theorem X.1.8 of [Rosser] p. 276. (Contributed by SF, 16-Jan-2015.) |
| Ref | Expression |
|---|---|
| addcid1 |
    0c 
|
are mutually distinct and distinct from all other
variables.| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-0c 4377 |
. . 3
0c
   | |
| 2 | 1 | addceq2i 4387 |
. 2
0c
|
| 3 | 0ex 4110 |
. . . . . . 7
  | |
| 4 | ineq2 3451 |
. . . . . . . . . 10
  | |
| 5 | 4 | eqeq1d 2361 |
. . . . . . . . 9
 |
| 6 | uneq2 3412 |
. . . . . . . . . 10
 | |
| 7 | 6 | eqeq2d 2364 |
. . . . . . . . 9
|
| 8 | 5, 7 | anbi12d 691 |
. . . . . . . 8
![](wedge.gif "/\"){kind=small}
|
| 9 | in0 3576 |
. . . . . . . . 9
| |
| 10 | 9 | biantrur 492 |
. . . . . . . 8
|
| 11 | 8, 10 | syl6bbr 254 |
. . . . . . 7
|
| 12 | 3, 11 | rexsn 3768 |
. . . . . 6
|
| 13 | un0 3575 |
. . . . . . 7
| |
| 14 | 13 | eqeq2i 2363 |
. . . . . 6
|
| 15 | equcom 1680 |
. . . . . 6
| |
| 16 | 12, 14, 15 | 3bitri 262 |
. . . . 5
|
| 17 | 16 | rexbii 2639 |
. . . 4
|
| 18 | eladdc 4398 |
. . . 4
| |
| 19 | risset 2661 |
. . . 4
| |
| 20 | 17, 18, 19 | 3bitr4i 268 |
. . 3
|
| 21 | 20 | eqriv 2350 |
. 2
|
| 22 | 2, 21 | eqtri 2373 |
1
0c
|
| Colors of variables: wff setvar class |
| Syntax hints: wa 358
wceq 1642
wcel 1710
wrex 2615
cun 3207
cin 3208
c0 3550
csn 3737
0cc0c 4374 cplc 4375 |
| This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-sn 4087 |
| This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-nul 3551 df-pw 3724 df-sn 3741 df-pr 3742 df-opk 4058 df-1c 4136 df-pw1 4137 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-sik 4192 df-ssetk 4193 df-0c 4377 df-addc 4378 |
| This theorem is referenced by: addcid2 4407 1cnnc 4408 nncaddccl 4419 ltfinirr 4457 ltfinp1 4462 lefinlteq 4463 lefinrflx 4467 vfin1cltv 4547 nclenn 6249 ncslesuc 6267 nncdiv3 6277 nnc3n3p1 6278 nchoicelem17 6305 |
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