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Mirrors > Home > MPE Home > Th. List > difsnexi | Structured version Visualization version GIF version |
Description: If the difference of a class and a singleton is a set, the class itself is a set. (Contributed by AV, 15-Jan-2019.) |
Ref | Expression |
---|---|
difsnexi | ⊢ ((𝑁 ∖ {𝐾}) ∈ V → 𝑁 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 479 | . . . . 5 ⊢ ((𝐾 ∈ 𝑁 ∧ (𝑁 ∖ {𝐾}) ∈ V) → (𝑁 ∖ {𝐾}) ∈ V) | |
2 | snex 5140 | . . . . 5 ⊢ {𝐾} ∈ V | |
3 | unexg 7236 | . . . . 5 ⊢ (((𝑁 ∖ {𝐾}) ∈ V ∧ {𝐾} ∈ V) → ((𝑁 ∖ {𝐾}) ∪ {𝐾}) ∈ V) | |
4 | 1, 2, 3 | sylancl 580 | . . . 4 ⊢ ((𝐾 ∈ 𝑁 ∧ (𝑁 ∖ {𝐾}) ∈ V) → ((𝑁 ∖ {𝐾}) ∪ {𝐾}) ∈ V) |
5 | difsnid 4572 | . . . . . . 7 ⊢ (𝐾 ∈ 𝑁 → ((𝑁 ∖ {𝐾}) ∪ {𝐾}) = 𝑁) | |
6 | 5 | eqcomd 2784 | . . . . . 6 ⊢ (𝐾 ∈ 𝑁 → 𝑁 = ((𝑁 ∖ {𝐾}) ∪ {𝐾})) |
7 | 6 | eleq1d 2844 | . . . . 5 ⊢ (𝐾 ∈ 𝑁 → (𝑁 ∈ V ↔ ((𝑁 ∖ {𝐾}) ∪ {𝐾}) ∈ V)) |
8 | 7 | adantr 474 | . . . 4 ⊢ ((𝐾 ∈ 𝑁 ∧ (𝑁 ∖ {𝐾}) ∈ V) → (𝑁 ∈ V ↔ ((𝑁 ∖ {𝐾}) ∪ {𝐾}) ∈ V)) |
9 | 4, 8 | mpbird 249 | . . 3 ⊢ ((𝐾 ∈ 𝑁 ∧ (𝑁 ∖ {𝐾}) ∈ V) → 𝑁 ∈ V) |
10 | 9 | ex 403 | . 2 ⊢ (𝐾 ∈ 𝑁 → ((𝑁 ∖ {𝐾}) ∈ V → 𝑁 ∈ V)) |
11 | difsn 4560 | . . . 4 ⊢ (¬ 𝐾 ∈ 𝑁 → (𝑁 ∖ {𝐾}) = 𝑁) | |
12 | 11 | eleq1d 2844 | . . 3 ⊢ (¬ 𝐾 ∈ 𝑁 → ((𝑁 ∖ {𝐾}) ∈ V ↔ 𝑁 ∈ V)) |
13 | 12 | biimpd 221 | . 2 ⊢ (¬ 𝐾 ∈ 𝑁 → ((𝑁 ∖ {𝐾}) ∈ V → 𝑁 ∈ V)) |
14 | 10, 13 | pm2.61i 177 | 1 ⊢ ((𝑁 ∖ {𝐾}) ∈ V → 𝑁 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 198 ∧ wa 386 ∈ wcel 2107 Vcvv 3398 ∖ cdif 3789 ∪ cun 3790 {csn 4398 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-sn 4399 df-pr 4401 df-uni 4672 |
This theorem is referenced by: pmtrdifellem1 18279 pmtrdifellem2 18280 tgdif0 21204 |
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