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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 empty. (Contributed by NM, 18-Aug-2018.) |
Ref | Expression |
---|---|
csb0 | ⊢ ⦋𝐴 / 𝑥⦌∅ = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | csbconstg 3512 | . 2 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌∅ = ∅) | |
2 | csbprc 3932 | . 2 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌∅ = ∅) | |
3 | 1, 2 | pm2.61i 175 | 1 ⊢ ⦋𝐴 / 𝑥⦌∅ = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 Vcvv 3173 ⦋csb 3499 ∅c0 3874 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-nul 3875 |
This theorem is referenced by: disjdsct 28863 onfrALTlem5 37778 onfrALTlem4 37779 |
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