![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > difexd | Structured version Visualization version GIF version |
Description: Existence of a difference. (Contributed by SN, 16-Jul-2024.) |
Ref | Expression |
---|---|
difexd.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
Ref | Expression |
---|---|
difexd | ⊢ (𝜑 → (𝐴 ∖ 𝐵) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difexd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | difexg 5190 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∖ 𝐵) ∈ V) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → (𝐴 ∖ 𝐵) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2112 Vcvv 3407 ∖ cdif 3851 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-ext 2730 ax-sep 5162 |
This theorem depends on definitions: df-bi 210 df-an 401 df-tru 1542 df-ex 1783 df-sb 2071 df-clab 2737 df-cleq 2751 df-clel 2831 df-rab 3077 df-v 3409 df-dif 3857 df-in 3861 df-ss 3871 |
This theorem is referenced by: hashf1lem1 13849 sexp2 33333 sexp3 33339 fsuppssind 39772 clcnvlem 40681 |
Copyright terms: Public domain | W3C validator |