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Mirrors > Home > MPE Home > Th. List > pssv | Structured version Visualization version GIF version |
Description: Any non-universal class is a proper subclass of the universal class. (Contributed by NM, 17-May-1998.) |
Ref | Expression |
---|---|
pssv | ⊢ (𝐴 ⊊ V ↔ ¬ 𝐴 = V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssv 3999 | . 2 ⊢ 𝐴 ⊆ V | |
2 | dfpss2 4078 | . 2 ⊢ (𝐴 ⊊ V ↔ (𝐴 ⊆ V ∧ ¬ 𝐴 = V)) | |
3 | 1, 2 | mpbiran 706 | 1 ⊢ (𝐴 ⊊ V ↔ ¬ 𝐴 = V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 = wceq 1533 Vcvv 3466 ⊆ wss 3941 ⊊ wpss 3942 |
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-ne 2933 df-v 3468 df-in 3948 df-ss 3958 df-pss 3960 |
This theorem is referenced by: (None) |
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