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Mirrors > Home > ILE Home > Th. List > ddifss | GIF version |
Description: Double complement under universal class. In classical logic (or given an additional hypothesis, as in ddifnel 3258), this is equality rather than subset. (Contributed by Jim Kingdon, 24-Jul-2018.) |
Ref | Expression |
---|---|
ddifss | ⊢ 𝐴 ⊆ (V ∖ (V ∖ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssv 3169 | . 2 ⊢ 𝐴 ⊆ V | |
2 | ssddif 3361 | . 2 ⊢ (𝐴 ⊆ V ↔ 𝐴 ⊆ (V ∖ (V ∖ 𝐴))) | |
3 | 1, 2 | mpbi 144 | 1 ⊢ 𝐴 ⊆ (V ∖ (V ∖ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: Vcvv 2730 ∖ cdif 3118 ⊆ wss 3121 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-v 2732 df-dif 3123 df-in 3127 df-ss 3134 |
This theorem is referenced by: ssindif0im 3473 difdifdirss 3498 |
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