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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 4002 | . 2 ⊢ 𝐴 ⊆ V | |
2 | dfpss2 4081 | . 2 ⊢ (𝐴 ⊊ V ↔ (𝐴 ⊆ V ∧ ¬ 𝐴 = V)) | |
3 | 1, 2 | mpbiran 708 | 1 ⊢ (𝐴 ⊊ V ↔ ¬ 𝐴 = V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 = wceq 1534 Vcvv 3470 ⊆ wss 3945 ⊊ wpss 3946 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2937 df-v 3472 df-in 3952 df-ss 3962 df-pss 3964 |
This theorem is referenced by: (None) |
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