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Mirrors > Home > ILE Home > Th. List > Mathboxes > bdnel | GIF version |
Description: Non-membership of a setvar in a bounded formula is a bounded formula. (Contributed by BJ, 16-Oct-2019.) |
Ref | Expression |
---|---|
bdnel.1 | ⊢ BOUNDED 𝐴 |
Ref | Expression |
---|---|
bdnel | ⊢ BOUNDED 𝑥 ∉ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bdnel.1 | . . . 4 ⊢ BOUNDED 𝐴 | |
2 | 1 | bdeli 15076 | . . 3 ⊢ BOUNDED 𝑥 ∈ 𝐴 |
3 | 2 | ax-bdn 15047 | . 2 ⊢ BOUNDED ¬ 𝑥 ∈ 𝐴 |
4 | df-nel 2456 | . 2 ⊢ (𝑥 ∉ 𝐴 ↔ ¬ 𝑥 ∈ 𝐴) | |
5 | 3, 4 | bd0r 15055 | 1 ⊢ BOUNDED 𝑥 ∉ 𝐴 |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∈ wcel 2160 ∉ wnel 2455 BOUNDED wbd 15042 BOUNDED wbdc 15070 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-4 1521 ax-bd0 15043 ax-bdn 15047 |
This theorem depends on definitions: df-bi 117 df-nel 2456 df-bdc 15071 |
This theorem is referenced by: (None) |
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