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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 3994 | . 2 ⊢ 𝐴 ⊆ V | |
2 | dfpss2 4065 | . 2 ⊢ (𝐴 ⊊ V ↔ (𝐴 ⊆ V ∧ ¬ 𝐴 = V)) | |
3 | 1, 2 | mpbiran 707 | 1 ⊢ (𝐴 ⊊ V ↔ ¬ 𝐴 = V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 = wceq 1536 Vcvv 3497 ⊆ wss 3939 ⊊ wpss 3940 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-ne 3020 df-v 3499 df-in 3946 df-ss 3955 df-pss 3957 |
This theorem is referenced by: (None) |
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