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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 3918 | . 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌∅ = ∅) | |
| 2 | csbprc 4409 | . 2 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌∅ = ∅) | |
| 3 | 1, 2 | pm2.61i 182 | 1 ⊢ ⦋𝐴 / 𝑥⦌∅ = ∅ | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1540 ∈ wcel 2108 Vcvv 3480 ⦋csb 3899 ∅c0 4333 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-nul 4334 | 
| This theorem is referenced by: disjdsct 32712 onfrALTlem5 44562 onfrALTlem4 44563 onfrALTlem5VD 44905 onfrALTlem4VD 44906 | 
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