New Foundations Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > NFE Home > Th. List > eldifsn | GIF version |
Description: Membership in a set with an element removed. (Contributed by NM, 10-Oct-2007.) |
Ref | Expression |
---|---|
eldifsn | ⊢ (A ∈ (B ∖ {C}) ↔ (A ∈ B ∧ A ≠ C)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldif 3221 | . 2 ⊢ (A ∈ (B ∖ {C}) ↔ (A ∈ B ∧ ¬ A ∈ {C})) | |
2 | elsncg 3755 | . . . 4 ⊢ (A ∈ B → (A ∈ {C} ↔ A = C)) | |
3 | 2 | necon3bbid 2550 | . . 3 ⊢ (A ∈ B → (¬ A ∈ {C} ↔ A ≠ C)) |
4 | 3 | pm5.32i 618 | . 2 ⊢ ((A ∈ B ∧ ¬ A ∈ {C}) ↔ (A ∈ B ∧ A ≠ C)) |
5 | 1, 4 | bitri 240 | 1 ⊢ (A ∈ (B ∖ {C}) ↔ (A ∈ B ∧ A ≠ C)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 176 ∧ wa 358 ∈ wcel 1710 ≠ wne 2516 ∖ cdif 3206 {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-sn 3741 |
This theorem is referenced by: eldifsni 3840 rexdifsn 3843 difsn 3845 nnsucelrlem2 4425 evenfinex 4503 oddfinex 4504 evenoddnnnul 4514 vfinspnn 4541 vfinspsslem1 4550 vinf 4555 enadjlem1 6059 2p1e3c 6156 |
Copyright terms: Public domain | W3C validator |