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Theorem bj-axempty 15048
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 4143. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 4144 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 15047 . 2 𝑥𝑦(𝑦𝑥 → ⊥)
2 df-ral 2473 . . 3 (∀𝑦𝑥 ⊥ ↔ ∀𝑦(𝑦𝑥 → ⊥))
32exbii 1616 . 2 (∃𝑥𝑦𝑥 ⊥ ↔ ∃𝑥𝑦(𝑦𝑥 → ⊥))
41, 3mpbir 146 1 𝑥𝑦𝑥
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1362  wfal 1369  wex 1503  wral 2468
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 1458  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-4 1521  ax-ial 1545  ax-bd0 14968  ax-bdim 14969  ax-bdn 14972  ax-bdeq 14975  ax-bdsep 15039
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-ral 2473
This theorem is referenced by: (None)
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