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Mirrors > Home > ILE Home > Th. List > difidALT | GIF version |
Description: The difference between a class and itself is the empty set. Proposition 5.15 of [TakeutiZaring] p. 20. Also Theorem 32 of [Suppes] p. 28. Alternate proof of difid 3491. (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 3137 | . 2 ⊢ (𝐴 ∖ 𝐴) = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐴} | |
2 | dfnul3 3425 | . 2 ⊢ ∅ = {𝑥 ∈ 𝐴 ∣ ¬ 𝑥 ∈ 𝐴} | |
3 | 1, 2 | eqtr4i 2201 | 1 ⊢ (𝐴 ∖ 𝐴) = ∅ |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 = wceq 1353 ∈ wcel 2148 {crab 2459 ∖ cdif 3126 ∅c0 3422 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-rab 2464 df-v 2739 df-dif 3131 df-nul 3423 |
This theorem is referenced by: (None) |
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