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Mirrors > Home > NFE Home > Th. List > nfnin | GIF version |
Description: Hypothesis builder for anti-intersection. (Contributed by SF, 2-Jan-2018.) |
Ref | Expression |
---|---|
nfnin.1 | ⊢ ℲxA |
nfnin.2 | ⊢ ℲxB |
Ref | Expression |
---|---|
nfnin | ⊢ Ⅎx(A ⩃ B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-nin 3211 | . 2 ⊢ (A ⩃ B) = {y ∣ (y ∈ A ⊼ y ∈ B)} | |
2 | nfnin.1 | . . . . 5 ⊢ ℲxA | |
3 | 2 | nfel2 2501 | . . . 4 ⊢ Ⅎx y ∈ A |
4 | nfnin.2 | . . . . 5 ⊢ ℲxB | |
5 | 4 | nfel2 2501 | . . . 4 ⊢ Ⅎx y ∈ B |
6 | 3, 5 | nfnan 1825 | . . 3 ⊢ Ⅎx(y ∈ A ⊼ y ∈ B) |
7 | 6 | nfab 2493 | . 2 ⊢ Ⅎx{y ∣ (y ∈ A ⊼ y ∈ B)} |
8 | 1, 7 | nfcxfr 2486 | 1 ⊢ Ⅎx(A ⩃ B) |
Colors of variables: wff setvar class |
Syntax hints: ⊼ wnan 1287 ∈ wcel 1710 {cab 2339 Ⅎwnfc 2476 ⩃ cnin 3204 |
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-nin 3211 |
This theorem is referenced by: nfcompl 3229 nfin 3230 nfun 3231 |
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