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Mirrors > Home > MPE Home > Th. List > difidALT | Structured version Visualization version GIF version |
Description: Alternate proof of difid 4330. (Contributed by David Abernethy, 17-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
difidALT | ⊢ (𝐴 ∖ 𝐴) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfdif2 3945 | . 2 ⊢ (𝐴 ∖ 𝐴) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐴} | |
2 | dfnul3 4295 | . 2 ⊢ ∅ = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐴} | |
3 | 1, 2 | eqtr4i 2847 | 1 ⊢ (𝐴 ∖ 𝐴) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1537 ∈ wcel 2114 {crab 3142 ∖ cdif 3933 ∅c0 4291 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-9 2124 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-sb 2070 df-clab 2800 df-cleq 2814 df-rab 3147 df-dif 3939 df-nul 4292 |
This theorem is referenced by: (None) |
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