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Theorem bj-axempty 13018
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 4023. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4024 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 13017 . 2 𝑥𝑦(𝑦𝑥 → ⊥)
2 df-ral 2398 . . 3 (∀𝑦𝑥 ⊥ ↔ ∀𝑦(𝑦𝑥 → ⊥))
32exbii 1569 . 2 (∃𝑥𝑦𝑥 ⊥ ↔ ∃𝑥𝑦(𝑦𝑥 → ⊥))
41, 3mpbir 145 1 𝑥𝑦𝑥
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1314  wfal 1321  wex 1453  wral 2393
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 588  ax-in2 589  ax-5 1408  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-4 1472  ax-ial 1499  ax-bd0 12938  ax-bdim 12939  ax-bdn 12942  ax-bdeq 12945  ax-bdsep 13009
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-fal 1322  df-ral 2398
This theorem is referenced by: (None)
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