![]() |
Mathbox for BJ |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > Mathboxes > bj-axempty | GIF version |
Description: Axiom of the empty set from bounded separation. It is provable from bounded separation since the intuitionistic FOL used in iset.mm assumes a nonempty universe. See axnul 4140. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4141 instead. (New usage is discouraged.) |
Ref | Expression |
---|---|
bj-axempty | ⊢ ∃𝑥∀𝑦 ∈ 𝑥 ⊥ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-axemptylem 14915 | . 2 ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 → ⊥) | |
2 | df-ral 2470 | . . 3 ⊢ (∀𝑦 ∈ 𝑥 ⊥ ↔ ∀𝑦(𝑦 ∈ 𝑥 → ⊥)) | |
3 | 2 | exbii 1615 | . 2 ⊢ (∃𝑥∀𝑦 ∈ 𝑥 ⊥ ↔ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 → ⊥)) |
4 | 1, 3 | mpbir 146 | 1 ⊢ ∃𝑥∀𝑦 ∈ 𝑥 ⊥ |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1361 ⊥wfal 1368 ∃wex 1502 ∀wral 2465 |
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-in1 615 ax-in2 616 ax-5 1457 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-4 1520 ax-ial 1544 ax-bd0 14836 ax-bdim 14837 ax-bdn 14840 ax-bdeq 14843 ax-bdsep 14907 |
This theorem depends on definitions: df-bi 117 df-tru 1366 df-fal 1369 df-ral 2470 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |