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Mirrors > Home > ILE Home > Th. List > dif0 | GIF version |
Description: The difference between a class and the empty set. Part of Exercise 4.4 of [Stoll] p. 16. (Contributed by NM, 17-Aug-2004.) |
Ref | Expression |
---|---|
dif0 | ⊢ (𝐴 ∖ ∅) = 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difid 3491 | . . 3 ⊢ (𝐴 ∖ 𝐴) = ∅ | |
2 | 1 | difeq2i 3250 | . 2 ⊢ (𝐴 ∖ (𝐴 ∖ 𝐴)) = (𝐴 ∖ ∅) |
3 | difdif 3260 | . 2 ⊢ (𝐴 ∖ (𝐴 ∖ 𝐴)) = 𝐴 | |
4 | 2, 3 | eqtr3i 2200 | 1 ⊢ (𝐴 ∖ ∅) = 𝐴 |
Colors of variables: wff set class |
Syntax hints: = wceq 1353 ∖ 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-ral 2460 df-rab 2464 df-v 2739 df-dif 3131 df-in 3135 df-ss 3142 df-nul 3423 |
This theorem is referenced by: disjdif2 3501 2oconcl 6433 diffifi 6887 undifdc 6916 difinfinf 7093 ismkvnex 7146 0cld 13245 exmid1stab 14372 |
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