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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 3904 | . 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌∅ = ∅) | |
2 | csbprc 4360 | . 2 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌∅ = ∅) | |
3 | 1, 2 | pm2.61i 184 | 1 ⊢ ⦋𝐴 / 𝑥⦌∅ = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 Vcvv 3496 ⦋csb 3885 ∅c0 4293 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-nul 4294 |
This theorem is referenced by: disjdsct 30440 onfrALTlem5 40883 onfrALTlem4 40884 onfrALTlem5VD 41226 onfrALTlem4VD 41227 |
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