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Mirrors > Home > MPE Home > Th. List > difidALT | Structured version Visualization version GIF version |
Description: Alternate proof of difid 4331. Shorter, but requiring ax-8 2109, df-clel 2811. (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 3967 | . 2 ⊢ 𝐴 ⊆ 𝐴 | |
2 | ssdif0 4324 | . 2 ⊢ (𝐴 ⊆ 𝐴 ↔ (𝐴 ∖ 𝐴) = ∅) | |
3 | 1, 2 | mpbi 229 | 1 ⊢ (𝐴 ∖ 𝐴) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∖ cdif 3908 ⊆ wss 3911 ∅c0 4283 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3446 df-dif 3914 df-in 3918 df-ss 3928 df-nul 4284 |
This theorem is referenced by: (None) |
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