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Mirrors > Home > MPE Home > Th. List > class2set | Structured version Visualization version GIF version |
Description: Construct, from any class 𝐴, a set equal to it when the class exists and equal to the empty set when the class is proper. This theorem shows that the constructed set always exists. (Contributed by NM, 16-Oct-2003.) |
Ref | Expression |
---|---|
class2set | ⊢ {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rabexg 5198 | . 2 ⊢ (𝐴 ∈ V → {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} ∈ V) | |
2 | simpl 486 | . . . . 5 ⊢ ((¬ 𝐴 ∈ V ∧ 𝑥 ∈ 𝐴) → ¬ 𝐴 ∈ V) | |
3 | 2 | nrexdv 3229 | . . . 4 ⊢ (¬ 𝐴 ∈ V → ¬ ∃𝑥 ∈ 𝐴 𝐴 ∈ V) |
4 | rabn0 4293 | . . . . 5 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 𝐴 ∈ V) | |
5 | 4 | necon1bbii 3036 | . . . 4 ⊢ (¬ ∃𝑥 ∈ 𝐴 𝐴 ∈ V ↔ {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} = ∅) |
6 | 3, 5 | sylib 221 | . . 3 ⊢ (¬ 𝐴 ∈ V → {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} = ∅) |
7 | 0ex 5175 | . . 3 ⊢ ∅ ∈ V | |
8 | 6, 7 | eqeltrdi 2898 | . 2 ⊢ (¬ 𝐴 ∈ V → {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} ∈ V) |
9 | 1, 8 | pm2.61i 185 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 {crab 3110 Vcvv 3441 ∅c0 4243 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-dif 3884 df-in 3888 df-ss 3898 df-nul 4244 |
This theorem is referenced by: (None) |
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