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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 3625 | . 2 ⊢ 𝐴 ⊆ V | |
2 | dfpss2 3692 | . 2 ⊢ (𝐴 ⊊ V ↔ (𝐴 ⊆ V ∧ ¬ 𝐴 = V)) | |
3 | 1, 2 | mpbiran 953 | 1 ⊢ (𝐴 ⊊ V ↔ ¬ 𝐴 = V) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 196 = wceq 1483 Vcvv 3200 ⊆ wss 3574 ⊊ wpss 3575 |
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 |
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-ne 2795 df-v 3202 df-in 3581 df-ss 3588 df-pss 3590 |
This theorem is referenced by: (None) |
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