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Mirrors > Home > NFE Home > Th. List > difsnpss | GIF version |
Description: (B ∖ {A}) is a proper subclass of B if and only if A is a member of B. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
difsnpss | ⊢ (A ∈ B ↔ (B ∖ {A}) ⊊ B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | notnot 282 | . 2 ⊢ (A ∈ B ↔ ¬ ¬ A ∈ B) | |
2 | difss 3393 | . . . 4 ⊢ (B ∖ {A}) ⊆ B | |
3 | 2 | biantrur 492 | . . 3 ⊢ ((B ∖ {A}) ≠ B ↔ ((B ∖ {A}) ⊆ B ∧ (B ∖ {A}) ≠ B)) |
4 | difsnb 3850 | . . . 4 ⊢ (¬ A ∈ B ↔ (B ∖ {A}) = B) | |
5 | 4 | necon3bbii 2547 | . . 3 ⊢ (¬ ¬ A ∈ B ↔ (B ∖ {A}) ≠ B) |
6 | df-pss 3261 | . . 3 ⊢ ((B ∖ {A}) ⊊ B ↔ ((B ∖ {A}) ⊆ B ∧ (B ∖ {A}) ≠ B)) | |
7 | 3, 5, 6 | 3bitr4i 268 | . 2 ⊢ (¬ ¬ A ∈ B ↔ (B ∖ {A}) ⊊ B) |
8 | 1, 7 | bitri 240 | 1 ⊢ (A ∈ B ↔ (B ∖ {A}) ⊊ B) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 176 ∧ wa 358 ∈ wcel 1710 ≠ wne 2516 ∖ cdif 3206 ⊆ wss 3257 ⊊ wpss 3258 {csn 3737 |
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-ne 2518 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-dif 3215 df-ss 3259 df-pss 3261 df-sn 3741 |
This theorem is referenced by: (None) |
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