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Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-axempty2 | GIF version |
Description: Axiom of the empty set from bounded separation, alternate version to bj-axempty 13928. (Contributed by BJ, 27-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4115 instead. (New usage is discouraged.) |
Ref | Expression |
---|---|
bj-axempty2 | ⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-axemptylem 13927 | . 2 ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 → ⊥) | |
2 | dfnot 1366 | . . . 4 ⊢ (¬ 𝑦 ∈ 𝑥 ↔ (𝑦 ∈ 𝑥 → ⊥)) | |
3 | 2 | albii 1463 | . . 3 ⊢ (∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∀𝑦(𝑦 ∈ 𝑥 → ⊥)) |
4 | 3 | exbii 1598 | . 2 ⊢ (∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 ↔ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 → ⊥)) |
5 | 1, 4 | mpbir 145 | 1 ⊢ ∃𝑥∀𝑦 ¬ 𝑦 ∈ 𝑥 |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1346 ⊥wfal 1353 ∃wex 1485 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-5 1440 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-4 1503 ax-ial 1527 ax-bd0 13848 ax-bdim 13849 ax-bdn 13852 ax-bdeq 13855 ax-bdsep 13919 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-fal 1354 |
This theorem is referenced by: (None) |
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