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Theorem bj-axempty 13260
 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 4060. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4061 instead. (New usage is discouraged.)
Assertion
Ref Expression
bj-axempty 𝑥𝑦𝑥
Distinct variable group:   𝑥,𝑦

Proof of Theorem bj-axempty
StepHypRef Expression
1 bj-axemptylem 13259 . 2 𝑥𝑦(𝑦𝑥 → ⊥)
2 df-ral 2422 . . 3 (∀𝑦𝑥 ⊥ ↔ ∀𝑦(𝑦𝑥 → ⊥))
32exbii 1585 . 2 (∃𝑥𝑦𝑥 ⊥ ↔ ∃𝑥𝑦(𝑦𝑥 → ⊥))
41, 3mpbir 145 1 𝑥𝑦𝑥
 Colors of variables: wff set class Syntax hints:   → wi 4  ∀wal 1330  ⊥wfal 1337  ∃wex 1469  ∀wral 2417 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 604  ax-in2 605  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-4 1488  ax-ial 1515  ax-bd0 13180  ax-bdim 13181  ax-bdn 13184  ax-bdeq 13187  ax-bdsep 13251 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338  df-ral 2422 This theorem is referenced by: (None)
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