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Mirrors > Home > NFE Home > Th. List > notrab | GIF version |
Description: Complementation of restricted class abstractions. (Contributed by Mario Carneiro, 3-Sep-2015.) |
Ref | Expression |
---|---|
notrab | ⊢ (A ∖ {x ∈ A ∣ φ}) = {x ∈ A ∣ ¬ φ} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difab 3523 | . 2 ⊢ ({x ∣ x ∈ A} ∖ {x ∣ φ}) = {x ∣ (x ∈ A ∧ ¬ φ)} | |
2 | difin 3492 | . . 3 ⊢ (A ∖ (A ∩ {x ∣ φ})) = (A ∖ {x ∣ φ}) | |
3 | dfrab3 3531 | . . . 4 ⊢ {x ∈ A ∣ φ} = (A ∩ {x ∣ φ}) | |
4 | 3 | difeq2i 3382 | . . 3 ⊢ (A ∖ {x ∈ A ∣ φ}) = (A ∖ (A ∩ {x ∣ φ})) |
5 | abid2 2470 | . . . 4 ⊢ {x ∣ x ∈ A} = A | |
6 | 5 | difeq1i 3381 | . . 3 ⊢ ({x ∣ x ∈ A} ∖ {x ∣ φ}) = (A ∖ {x ∣ φ}) |
7 | 2, 4, 6 | 3eqtr4i 2383 | . 2 ⊢ (A ∖ {x ∈ A ∣ φ}) = ({x ∣ x ∈ A} ∖ {x ∣ φ}) |
8 | df-rab 2623 | . 2 ⊢ {x ∈ A ∣ ¬ φ} = {x ∣ (x ∈ A ∧ ¬ φ)} | |
9 | 1, 7, 8 | 3eqtr4i 2383 | 1 ⊢ (A ∖ {x ∈ A ∣ φ}) = {x ∈ A ∣ ¬ φ} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 358 = wceq 1642 ∈ wcel 1710 {cab 2339 {crab 2618 ∖ cdif 3206 ∩ cin 3208 |
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-rab 2623 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 df-dif 3215 |
This theorem is referenced by: (None) |
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