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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 5216 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ V) → 𝐴 ∈ V) | |
2 | 1 | ex 416 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐵 ∈ V → 𝐴 ∈ V)) |
3 | 2 | nelcon3d 3058 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∉ V → 𝐵 ∉ V)) |
4 | 3 | imp 410 | 1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ∉ V) → 𝐵 ∉ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2110 ∉ wnel 3046 Vcvv 3408 ⊆ wss 3866 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 ax-sep 5192 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-nel 3047 df-rab 3070 df-v 3410 df-in 3873 df-ss 3883 |
This theorem is referenced by: usgrprc 27354 rgrusgrprc 27677 rgrprc 27679 fsetprcnexALT 44228 |
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