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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 4812 | . 2 ⊢ (𝐴 ∈ V → {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} ∈ V) | |
2 | simpl 473 | . . . . 5 ⊢ ((¬ 𝐴 ∈ V ∧ 𝑥 ∈ 𝐴) → ¬ 𝐴 ∈ V) | |
3 | 2 | nrexdv 3001 | . . . 4 ⊢ (¬ 𝐴 ∈ V → ¬ ∃𝑥 ∈ 𝐴 𝐴 ∈ V) |
4 | rabn0 3958 | . . . . 5 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 𝐴 ∈ V) | |
5 | 4 | necon1bbii 2843 | . . . 4 ⊢ (¬ ∃𝑥 ∈ 𝐴 𝐴 ∈ V ↔ {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} = ∅) |
6 | 3, 5 | sylib 208 | . . 3 ⊢ (¬ 𝐴 ∈ V → {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} = ∅) |
7 | 0ex 4790 | . . 3 ⊢ ∅ ∈ V | |
8 | 6, 7 | syl6eqel 2709 | . 2 ⊢ (¬ 𝐴 ∈ V → {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} ∈ V) |
9 | 1, 8 | pm2.61i 176 | 1 ⊢ {𝑥 ∈ 𝐴 ∣ 𝐴 ∈ V} ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1483 ∈ wcel 1990 ∃wrex 2913 {crab 2916 Vcvv 3200 ∅c0 3915 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-in 3581 df-ss 3588 df-nul 3916 |
This theorem is referenced by: (None) |
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