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Mirrors > Home > ILE Home > Th. List > fidifsnid | GIF 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 3531 from subset to equality when the set is finite. (Contributed by Jim Kingdon, 9-Sep-2021.) |
Ref | Expression |
---|---|
fidifsnid | ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴) → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difsnss 3531 | . . 3 ⊢ (𝐵 ∈ 𝐴 → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) ⊆ 𝐴) | |
2 | 1 | adantl 271 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴) → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) ⊆ 𝐴) |
3 | simpr 108 | . . . . . . 7 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 = 𝐵) → 𝑥 = 𝐵) | |
4 | velsn 3415 | . . . . . . 7 ⊢ (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵) | |
5 | 3, 4 | sylibr 132 | . . . . . 6 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 = 𝐵) → 𝑥 ∈ {𝐵}) |
6 | elun2 3140 | . . . . . 6 ⊢ (𝑥 ∈ {𝐵} → 𝑥 ∈ ((𝐴 ∖ {𝐵}) ∪ {𝐵})) | |
7 | 5, 6 | syl 14 | . . . . 5 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) ∧ 𝑥 = 𝐵) → 𝑥 ∈ ((𝐴 ∖ {𝐵}) ∪ {𝐵})) |
8 | simplr 496 | . . . . . . 7 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ 𝐴) | |
9 | simpr 108 | . . . . . . . 8 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) ∧ ¬ 𝑥 = 𝐵) → ¬ 𝑥 = 𝐵) | |
10 | 9, 4 | sylnibr 634 | . . . . . . 7 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) ∧ ¬ 𝑥 = 𝐵) → ¬ 𝑥 ∈ {𝐵}) |
11 | 8, 10 | eldifd 2983 | . . . . . 6 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ (𝐴 ∖ {𝐵})) |
12 | elun1 3139 | . . . . . 6 ⊢ (𝑥 ∈ (𝐴 ∖ {𝐵}) → 𝑥 ∈ ((𝐴 ∖ {𝐵}) ∪ {𝐵})) | |
13 | 11, 12 | syl 14 | . . . . 5 ⊢ ((((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) ∧ ¬ 𝑥 = 𝐵) → 𝑥 ∈ ((𝐴 ∖ {𝐵}) ∪ {𝐵})) |
14 | simpll 495 | . . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → 𝐴 ∈ Fin) | |
15 | simpr 108 | . . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
16 | simplr 496 | . . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐴) | |
17 | fidceq 6354 | . . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ 𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴) → DECID 𝑥 = 𝐵) | |
18 | 14, 15, 16, 17 | syl3anc 1169 | . . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → DECID 𝑥 = 𝐵) |
19 | df-dc 776 | . . . . . 6 ⊢ (DECID 𝑥 = 𝐵 ↔ (𝑥 = 𝐵 ∨ ¬ 𝑥 = 𝐵)) | |
20 | 18, 19 | sylib 120 | . . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → (𝑥 = 𝐵 ∨ ¬ 𝑥 = 𝐵)) |
21 | 7, 13, 20 | mpjaodan 744 | . . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴) ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ((𝐴 ∖ {𝐵}) ∪ {𝐵})) |
22 | 21 | ex 113 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴) → (𝑥 ∈ 𝐴 → 𝑥 ∈ ((𝐴 ∖ {𝐵}) ∪ {𝐵}))) |
23 | 22 | ssrdv 3005 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴) → 𝐴 ⊆ ((𝐴 ∖ {𝐵}) ∪ {𝐵})) |
24 | 2, 23 | eqssd 3016 | 1 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ 𝐴) → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 102 ∨ wo 661 DECID wdc 775 = wceq 1284 ∈ wcel 1433 ∖ cdif 2970 ∪ cun 2971 ⊆ wss 2973 {csn 3398 Fincfn 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 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-dc 776 df-3or 920 df-3an 921 df-tru 1287 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-sbc 2816 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-int 3637 df-br 3786 df-opab 3840 df-tr 3876 df-id 4048 df-iord 4121 df-on 4123 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-en 6245 df-fin 6247 |
This theorem is referenced by: findcard2 6373 findcard2s 6374 |
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