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Mirrors > Home > NFE Home > Th. List > ssindif0 | GIF version |
Description: Subclass expressed in terms of intersection with difference from the universal class. (Contributed by NM, 17-Sep-2003.) |
Ref | Expression |
---|---|
ssindif0 | ⊢ (A ⊆ B ↔ (A ∩ (V ∖ B)) = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | disj2 3598 | . 2 ⊢ ((A ∩ (V ∖ B)) = ∅ ↔ A ⊆ (V ∖ (V ∖ B))) | |
2 | ddif 3398 | . . 3 ⊢ (V ∖ (V ∖ B)) = B | |
3 | 2 | sseq2i 3296 | . 2 ⊢ (A ⊆ (V ∖ (V ∖ B)) ↔ A ⊆ B) |
4 | 1, 3 | bitr2i 241 | 1 ⊢ (A ⊆ B ↔ (A ∩ (V ∖ B)) = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 = wceq 1642 Vcvv 2859 ∖ cdif 3206 ∩ cin 3208 ⊆ wss 3257 ∅c0 3550 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 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-ne 2518 df-ral 2619 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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