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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 3844 | . 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌∅ = ∅) | |
2 | csbprc 4335 | . 2 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌∅ = ∅) | |
3 | 1, 2 | pm2.61i 185 | 1 ⊢ ⦋𝐴 / 𝑥⦌∅ = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∈ wcel 2111 Vcvv 3420 ⦋csb 3825 ∅c0 4251 |
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 2113 ax-9 2121 ax-ext 2709 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2072 df-clab 2716 df-cleq 2730 df-clel 2817 df-v 3422 df-sbc 3709 df-csb 3826 df-dif 3883 df-nul 4252 |
This theorem is referenced by: disjdsct 30779 onfrALTlem5 41863 onfrALTlem4 41864 onfrALTlem5VD 42206 onfrALTlem4VD 42207 |
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