Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > prcssprc | Structured version Visualization version GIF version |
Description: The superclass of a proper class is a proper class. (Contributed by AV, 27-Dec-2020.) |
Ref | Expression |
---|---|
prcssprc | ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ∉ V) → 𝐵 ∉ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssexg 5218 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ V) → 𝐴 ∈ V) | |
2 | 1 | ex 413 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ∈ V → 𝐴 ∈ V)) |
3 | 2 | nelcon3d 3132 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∉ V → 𝐵 ∉ V)) |
4 | 3 | imp 407 | 1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ∉ V) → 𝐵 ∉ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2105 ∉ wnel 3120 Vcvv 3492 ⊆ wss 3933 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-nel 3121 df-rab 3144 df-v 3494 df-in 3940 df-ss 3949 |
This theorem is referenced by: usgrprc 26975 rgrusgrprc 27298 rgrprc 27300 |
Copyright terms: Public domain | W3C validator |