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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 487 | . . . . 5 ⊢ ((𝐾 ∈ 𝑁 ∧ (𝑁 ∖ {𝐾}) ∈ V) → (𝑁 ∖ {𝐾}) ∈ V) | |
2 | snex 5334 | . . . . 5 ⊢ {𝐾} ∈ V | |
3 | unexg 7474 | . . . . 5 ⊢ (((𝑁 ∖ {𝐾}) ∈ V ∧ {𝐾} ∈ V) → ((𝑁 ∖ {𝐾}) ∪ {𝐾}) ∈ V) | |
4 | 1, 2, 3 | sylancl 588 | . . . 4 ⊢ ((𝐾 ∈ 𝑁 ∧ (𝑁 ∖ {𝐾}) ∈ V) → ((𝑁 ∖ {𝐾}) ∪ {𝐾}) ∈ V) |
5 | difsnid 4745 | . . . . . . 7 ⊢ (𝐾 ∈ 𝑁 → ((𝑁 ∖ {𝐾}) ∪ {𝐾}) = 𝑁) | |
6 | 5 | eqcomd 2829 | . . . . . 6 ⊢ (𝐾 ∈ 𝑁 → 𝑁 = ((𝑁 ∖ {𝐾}) ∪ {𝐾})) |
7 | 6 | eleq1d 2899 | . . . . 5 ⊢ (𝐾 ∈ 𝑁 → (𝑁 ∈ V ↔ ((𝑁 ∖ {𝐾}) ∪ {𝐾}) ∈ V)) |
8 | 7 | adantr 483 | . . . 4 ⊢ ((𝐾 ∈ 𝑁 ∧ (𝑁 ∖ {𝐾}) ∈ V) → (𝑁 ∈ V ↔ ((𝑁 ∖ {𝐾}) ∪ {𝐾}) ∈ V)) |
9 | 4, 8 | mpbird 259 | . . 3 ⊢ ((𝐾 ∈ 𝑁 ∧ (𝑁 ∖ {𝐾}) ∈ V) → 𝑁 ∈ V) |
10 | 9 | ex 415 | . 2 ⊢ (𝐾 ∈ 𝑁 → ((𝑁 ∖ {𝐾}) ∈ V → 𝑁 ∈ V)) |
11 | difsn 4733 | . . . 4 ⊢ (¬ 𝐾 ∈ 𝑁 → (𝑁 ∖ {𝐾}) = 𝑁) | |
12 | 11 | eleq1d 2899 | . . 3 ⊢ (¬ 𝐾 ∈ 𝑁 → ((𝑁 ∖ {𝐾}) ∈ V ↔ 𝑁 ∈ V)) |
13 | 12 | biimpd 231 | . 2 ⊢ (¬ 𝐾 ∈ 𝑁 → ((𝑁 ∖ {𝐾}) ∈ V → 𝑁 ∈ V)) |
14 | 10, 13 | pm2.61i 184 | 1 ⊢ ((𝑁 ∖ {𝐾}) ∈ V → 𝑁 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2114 Vcvv 3496 ∖ cdif 3935 ∪ cun 3936 {csn 4569 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-sn 4570 df-pr 4572 df-uni 4841 |
This theorem is referenced by: pmtrdifellem1 18606 pmtrdifellem2 18607 tgdif0 21602 |
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