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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 521 | . . 3 | |
2 | en2sn 6700 | . . . 4 | |
3 | 2 | ad2ant2lr 501 | . . 3 |
4 | simplr 519 | . . . . 5 | |
5 | 4 | eldifbd 3078 | . . . 4 |
6 | disjsn 3580 | . . . 4 | |
7 | 5, 6 | sylibr 133 | . . 3 |
8 | elirr 4451 | . . . . 5 | |
9 | disjsn 3580 | . . . . 5 | |
10 | 8, 9 | mpbir 145 | . . . 4 |
11 | 10 | a1i 9 | . . 3 |
12 | unen 6703 | . . 3 | |
13 | 1, 3, 7, 11, 12 | syl22anc 1217 | . 2 |
14 | df-suc 4288 | . 2 | |
15 | 13, 14 | breqtrrdi 3965 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wceq 1331 wcel 1480 cvv 2681 cdif 3063 cun 3064 cin 3065 c0 3358 csn 3522 class class class wbr 3924 csuc 4282 com 4499 cen 6625 cfn 6627 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-id 4210 df-suc 4288 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-1o 6306 df-er 6422 df-en 6628 |
This theorem is referenced by: php5fin 6769 hashunlem 10543 |
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