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Theorem bj-axempty 10379
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 non-empty universe. See axnul 3909. (Contributed by BJ, 25-Oct-2020.) (Proof modification is discouraged.) Use ax-nul 3910 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 10378 . 2 𝑥𝑦(𝑦𝑥 → ⊥)
2 df-ral 2328 . . 3 (∀𝑦𝑥 ⊥ ↔ ∀𝑦(𝑦𝑥 → ⊥))
32exbii 1512 . 2 (∃𝑥𝑦𝑥 ⊥ ↔ ∃𝑥𝑦(𝑦𝑥 → ⊥))
41, 3mpbir 138 1 𝑥𝑦𝑥
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1257  wfal 1264  wex 1397  wral 2323
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-ial 1443  ax-bd0 10299  ax-bdim 10300  ax-bdn 10303  ax-bdeq 10306  ax-bdsep 10370
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-fal 1265  df-ral 2328
This theorem is referenced by: (None)
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