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Mirrors > Home > MPE Home > Th. List > difidALT | Structured version Visualization version GIF version |
Description: Alternate proof of difid 4285. Shorter, but requiring ax-8 2112, df-clel 2816. (Contributed by NM, 22-Apr-2004.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
difidALT | ⊢ (𝐴 ∖ 𝐴) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssid 3923 | . 2 ⊢ 𝐴 ⊆ 𝐴 | |
2 | ssdif0 4278 | . 2 ⊢ (𝐴 ⊆ 𝐴 ↔ (𝐴 ∖ 𝐴) = ∅) | |
3 | 1, 2 | mpbi 233 | 1 ⊢ (𝐴 ∖ 𝐴) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∖ cdif 3863 ⊆ wss 3866 ∅c0 4237 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3410 df-dif 3869 df-in 3873 df-ss 3883 df-nul 4238 |
This theorem is referenced by: (None) |
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