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Mirrors > Home > MPE Home > Th. List > vss | Structured version Visualization version GIF version |
Description: Only the universal class has the universal class as a subclass. (Contributed by NM, 17-Sep-2003.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Ref | Expression |
---|---|
vss | ⊢ (V ⊆ 𝐴 ↔ 𝐴 = V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssv 3998 | . . 3 ⊢ 𝐴 ⊆ V | |
2 | 1 | biantrur 530 | . 2 ⊢ (V ⊆ 𝐴 ↔ (𝐴 ⊆ V ∧ V ⊆ 𝐴)) |
3 | eqss 3989 | . 2 ⊢ (𝐴 = V ↔ (𝐴 ⊆ V ∧ V ⊆ 𝐴)) | |
4 | 2, 3 | bitr4i 278 | 1 ⊢ (V ⊆ 𝐴 ↔ 𝐴 = V) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 = wceq 1533 Vcvv 3466 ⊆ wss 3940 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-v 3468 df-in 3947 df-ss 3957 |
This theorem is referenced by: vdif0 4460 |
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