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Mirrors > Home > ILE Home > Th. List > nndifsnid | GIF version |
Description: If we remove a single element from a natural number then put it back in, we end up with the original natural number. This strengthens difsnss 3713 from subset to equality but the proof relies on equality being decidable. (Contributed by Jim Kingdon, 31-Aug-2021.) |
Ref | Expression |
---|---|
nndifsnid | ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elnn 4577 | . . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐴 ∈ ω) → 𝑥 ∈ ω) | |
2 | 1 | expcom 115 | . . . . 5 ⊢ (𝐴 ∈ ω → (𝑥 ∈ 𝐴 → 𝑥 ∈ ω)) |
3 | elnn 4577 | . . . . . 6 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝐴 ∈ ω) → 𝑦 ∈ ω) | |
4 | 3 | expcom 115 | . . . . 5 ⊢ (𝐴 ∈ ω → (𝑦 ∈ 𝐴 → 𝑦 ∈ ω)) |
5 | 2, 4 | anim12d 333 | . . . 4 ⊢ (𝐴 ∈ ω → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 ∈ ω ∧ 𝑦 ∈ ω))) |
6 | nndceq 6458 | . . . 4 ⊢ ((𝑥 ∈ ω ∧ 𝑦 ∈ ω) → DECID 𝑥 = 𝑦) | |
7 | 5, 6 | syl6 33 | . . 3 ⊢ (𝐴 ∈ ω → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → DECID 𝑥 = 𝑦)) |
8 | 7 | ralrimivv 2545 | . 2 ⊢ (𝐴 ∈ ω → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦) |
9 | dcdifsnid 6463 | . 2 ⊢ ((∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 DECID 𝑥 = 𝑦 ∧ 𝐵 ∈ 𝐴) → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = 𝐴) | |
10 | 8, 9 | sylan 281 | 1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ 𝐴) → ((𝐴 ∖ {𝐵}) ∪ {𝐵}) = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 DECID wdc 824 = wceq 1342 ∈ wcel 2135 ∀wral 2442 ∖ cdif 3108 ∪ cun 3109 {csn 3570 ωcom 4561 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-nul 4102 ax-pow 4147 ax-pr 4181 ax-un 4405 ax-setind 4508 ax-iinf 4559 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-nul 3405 df-pw 3555 df-sn 3576 df-pr 3577 df-uni 3784 df-int 3819 df-tr 4075 df-iord 4338 df-on 4340 df-suc 4343 df-iom 4562 |
This theorem is referenced by: phplem2 6810 |
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