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Mirrors > Home > ILE Home > Th. List > fiunsnnn | Unicode version |
Description: Adding one element to a finite set which is equinumerous to a natural number. (Contributed by Jim Kingdon, 13-Sep-2021.) |
Ref | Expression |
---|---|
fiunsnnn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simprr 498 | . . 3 | |
2 | en2sn 6313 | . . . 4 | |
3 | 2 | ad2ant2lr 493 | . . 3 |
4 | simplr 496 | . . . . 5 | |
5 | 4 | eldifbd 2985 | . . . 4 |
6 | disjsn 3454 | . . . 4 | |
7 | 5, 6 | sylibr 132 | . . 3 |
8 | elirr 4284 | . . . . 5 | |
9 | disjsn 3454 | . . . . 5 | |
10 | 8, 9 | mpbir 144 | . . . 4 |
11 | 10 | a1i 9 | . . 3 |
12 | unen 6316 | . . 3 | |
13 | 1, 3, 7, 11, 12 | syl22anc 1170 | . 2 |
14 | df-suc 4126 | . 2 | |
15 | 13, 14 | syl6breqr 3825 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 102 wceq 1284 wcel 1433 cvv 2601 cdif 2970 cun 2971 cin 2972 c0 3251 csn 3398 class class class wbr 3785 csuc 4120 com 4331 cen 6242 cfn 6244 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-ral 2353 df-rex 2354 df-v 2603 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-id 4048 df-suc 4126 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-1o 6024 df-er 6129 df-en 6245 |
This theorem is referenced by: php5fin 6366 |
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