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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 4053. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4054 instead. (New usage is discouraged.) |
Ref | Expression |
---|---|
bj-axempty | ⊢ ∃𝑥∀𝑦 ∈ 𝑥 ⊥ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-axemptylem 13090 | . 2 ⊢ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 → ⊥) | |
2 | df-ral 2421 | . . 3 ⊢ (∀𝑦 ∈ 𝑥 ⊥ ↔ ∀𝑦(𝑦 ∈ 𝑥 → ⊥)) | |
3 | 2 | exbii 1584 | . 2 ⊢ (∃𝑥∀𝑦 ∈ 𝑥 ⊥ ↔ ∃𝑥∀𝑦(𝑦 ∈ 𝑥 → ⊥)) |
4 | 1, 3 | mpbir 145 | 1 ⊢ ∃𝑥∀𝑦 ∈ 𝑥 ⊥ |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∀wal 1329 ⊥wfal 1336 ∃wex 1468 ∀wral 2416 |
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 603 ax-in2 604 ax-5 1423 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-4 1487 ax-ial 1514 ax-bd0 13011 ax-bdim 13012 ax-bdn 13015 ax-bdeq 13018 ax-bdsep 13082 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-fal 1337 df-ral 2421 |
This theorem is referenced by: (None) |
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