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Mirrors > Home > MPE Home > Th. List > csb0 | Structured version Visualization version GIF version |
Description: The proper substitution of a class into the empty set is the empty set. (Contributed by NM, 18-Aug-2018.) |
Ref | Expression |
---|---|
csb0 | ⊢ ⦋𝐴 / 𝑥⦌∅ = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbconstg 3908 | . 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌∅ = ∅) | |
2 | csbprc 4408 | . 2 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌∅ = ∅) | |
3 | 1, 2 | pm2.61i 182 | 1 ⊢ ⦋𝐴 / 𝑥⦌∅ = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2098 Vcvv 3461 ⦋csb 3889 ∅c0 4322 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-nul 4323 |
This theorem is referenced by: disjdsct 32564 onfrALTlem5 44123 onfrALTlem4 44124 onfrALTlem5VD 44466 onfrALTlem4VD 44467 |
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