![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 3927 | . 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌∅ = ∅) | |
2 | csbprc 4415 | . 2 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌∅ = ∅) | |
3 | 1, 2 | pm2.61i 182 | 1 ⊢ ⦋𝐴 / 𝑥⦌∅ = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2106 Vcvv 3478 ⦋csb 3908 ∅c0 4339 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-nul 4340 |
This theorem is referenced by: disjdsct 32718 onfrALTlem5 44540 onfrALTlem4 44541 onfrALTlem5VD 44883 onfrALTlem4VD 44884 |
Copyright terms: Public domain | W3C validator |