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Mirrors > Home > ILE Home > Th. List > fidifsnid | Unicode version |
Description: If we remove a single element from a finite set then put it back in, we end up with the original finite set. This strengthens difsnss 3702 from subset to equality when the set is finite. (Contributed by Jim Kingdon, 9-Sep-2021.) |
Ref | Expression |
---|---|
fidifsnid |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fidceq 6814 | . . . 4 DECID | |
2 | 1 | 3expb 1186 | . . 3 DECID |
3 | 2 | ralrimivva 2539 | . 2 DECID |
4 | dcdifsnid 6451 | . 2 DECID | |
5 | 3, 4 | sylan 281 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 DECID wdc 820 wceq 1335 wcel 2128 wral 2435 cdif 3099 cun 3100 csn 3560 cfn 6685 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-iinf 4547 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-br 3966 df-opab 4026 df-tr 4063 df-id 4253 df-iord 4326 df-on 4328 df-suc 4331 df-iom 4550 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-fv 5178 df-en 6686 df-fin 6688 |
This theorem is referenced by: findcard2 6834 findcard2s 6835 xpfi 6874 fisseneq 6876 zfz1isolem1 10711 |
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