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Mirrors > Home > NFE Home > Th. List > 0dif | GIF version |
Description: The difference between the empty set and a class. Part of Exercise 4.4 of [Stoll] p. 16. (Contributed by NM, 17-Aug-2004.) |
Ref | Expression |
---|---|
0dif | ⊢ (∅ ∖ A) = ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difss 3393 | . 2 ⊢ (∅ ∖ A) ⊆ ∅ | |
2 | ss0 3581 | . 2 ⊢ ((∅ ∖ A) ⊆ ∅ → (∅ ∖ A) = ∅) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (∅ ∖ A) = ∅ |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1642 ∖ cdif 3206 ⊆ wss 3257 ∅c0 3550 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-dif 3215 df-ss 3259 df-nul 3551 |
This theorem is referenced by: (None) |
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